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

Abstract

Modular graph functions are $SL(2,\mathbb{Z})$-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus $\tau$. 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 ${\Gamma }_{\infty }\backslash PSL(2,\mathbb{Z})$. 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 $\tau \to -\bar{\tau }$ 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 $w\le 11$ the bound is saturated and we exhibit a basis for ${\mathfrak{A}}_w$.

88 pages (of which 32 pages are appendices); proof of convergence of Poincare series; example calculation of Petersson inner product; and Lemma 7.5 on the integral of cuspidal functions added in revised version