Modular graph functions and odd cuspidal functions. Fourier and Poincaré series

Abstract

Abstract Modular graph functions are SL(2, ℤ)-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus τ. For one-loop graphs they reduce to real analytic Eisenstein series. We obtain the Fourier series, including the constant and non-constant Fourier modes, of all two-loop modular graph functions, as well as their Poincaré series with respect to Γ∞\PSL(2, ℤ). The Fourier and Poincaré series provide the tools to compute the Petersson inner product of two-loop modular graph functions using Rankin-Selberg-Zagier methods. Modular graph functions which are odd under τ → − τ ¯ $$ \overline{\tau} $$ are cuspidal functions, with exponential decay near the cusp, and exist starting at two loops. Holomorphic subgraph reduction and the sieve algorithm, developed in earlier work, are used to give a lower bound on the dimension of the space A $$ \mathfrak{A} $$ w of odd two-loop modular graph functions of weight w. For w ≤ 11 the bound is saturated and we exhibit a basis for A $$ \mathfrak{A} $$ w .

Keywords

• Scattering Amplitudes
• Superstrings and Heterotic Strings