Modular graph functions and odd cuspidal functions. Fourier and Poincar\xe9 series

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 → − 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 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 mathfrak{A} w.