Rademacher expansion of modular integrals

TL;DR

This work provides a rigorous analytic framework for evaluating integrals of non-holomorphic modular functions over the torus fundamental domain by combining a two-variable Lorentzian contour (Lorentzian inversion) with a two-dimensional Rademacher expansion. The main novelty is a canonical decomposition into a holomorphically split Lorentzian integral followed by a contour deformation to Ford circles, yielding a convergent series over Lorentzian cusps expressed through Dedekind sums and the Rademacher function. The resulting formulas enable analytic access to string-one-loop quantities, including closed-form expressions for the bosonic string cosmological constant and the SO(16) × SO(16) string cosmological constant, and extend to rational CFTs and toroidal compactifications. The approach illuminates the cusp structure of modular integrals and provides efficient, highly accurate computations that compare favorably with direct numerical integration, with broad potential applications in string perturbation theory and AdS3 quantum gravity contexts.

Abstract

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.

Figures (8)

• Figure 1: The integration contour over the original (left) and modified (right) fundamental domain. On the left, we have $(\mathop{\text{Re}} x, \mathop{\text{Re}} y)= (\mathop{\text{Re}} \tau, \mathop{\text{Im}} \tau)$.
• Figure 2: The fundamental domain of $\Gamma(2)$.
• Figure 3: The Hankel contour $\mathcal{H}_\delta$ in the $v$-plane.
• Figure 4: Contours in the holomorphically factorized contour.
• Figure 5: Rademacher contour $\Gamma_\infty$ in the $\tilde{\tau}$-plane, enclosing Ford circles $C_{a/c}$ for all irreducible fractions $\frac{a}{c} \in [0,1)$.