Rademacher expansion of modular integrals

TL;DR

This paper introduces a novel method combining Lorentzian contour transformation and Rademacher expansion to analytically evaluate integrals of non-holomorphic modular functions, with applications to string theory partition functions.

Contribution

It develops a new technique for computing modular integrals analytically using a two-step process involving Lorentzian contours and Rademacher expansion, applicable to string theory.

Findings

First analytic formula for bosonic string cosmological constants

Application to string one-loop partition functions

Method sensitive to integrand singularities near Lorentzian cusps

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.