Rademacher expansion of modular integrals

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.

40 pages + appendices. v2 (references added)

References in corpus (25)

• Quantum Gravity Partition Functions in Three Dimensions
• Analyticity in Spin in Conformal Theories
• Light-ray operators in conformal field theory
• Causality Constraints in Conformal Field Theory
• A spacetime derivation of the Lorentzian OPE inversion formula
• Poincare Series, 3D Gravity and CFT Spectroscopy
• On the modular structure of the genus-one Type II superstring low energy expansion
• Nonperturbative black hole entropy and Kloosterman sums
• Harnessing S-Duality in $\mathcal{N}=4$ SYM & Supergravity as $SL(2,\mathbb{Z})$-Averaged Strings
• Harmonic analysis of 2d CFT partition functions
• Rademacher Expansions and the Spectrum of 2d CFT
• Unitarity Cuts of the Worldsheet
• Evaluating one-loop string amplitudes
• Symmetries and spectral statistics in chaotic conformal field theories
• S-duality in $T\bar{T}$-deformed CFT
• Scalar Modular Bootstrap and Zeros of the Riemann Zeta Function
• Black hole microstate counting from the gravitational path integral
• Renormalization and BRST symmetry in Donaldson-Witten theory
• The gravitational path integral for $N=4$ BPS black holes from black hole microstate counting
• One-loop Double Copy Relation in String Theory
• Rademacher expansion of a Siegel modular form for $\mathcal{N}=4$ counting
• The Feynman $i\epsilon$ in String Theory
• New Non-Supersymmetric Tachyon-Free Strings
• The $i\epsilon$-Prescription for String Amplitudes and Regularized Modular Integrals
• One-loop Amplitudes as BPS state sums

Cited by in corpus (2)

• One-loop four-graviton string amplitude at finite ${\alpha }^{\prime }$
• A glimpse into the Ultrametric spectrum