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

Abstract

A bstract 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 → − $$ \overline{\tau} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>τ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> 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 $$ \mathfrak{A} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> w of odd two-loop modular graph functions of weight w . For w ≤ 11 the bound is saturated and we exhibit a basis for $$ \mathfrak{A} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> w .