# Modular graph functions and odd cuspidal functions -- Fourier and Poincar\'e series

## TL;DR

This paper derives Fourier and Poincaré series for two-loop modular graph functions, introduces odd cuspidal functions starting at two loops, and constructs a basis for their space at low weights.

## Contribution

It provides explicit Fourier and Poincaré series for two-loop modular graph functions and characterizes odd cuspidal functions, expanding understanding of their structure and basis.

## Key findings

- Fourier series for all two-loop modular graph functions derived.
- Poincaré series enable Petersson inner product computation.
- A basis for odd two-loop modular graph functions established for weights up to 11.

## 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 Poincar\'e series with respect to $\Gamma_\infty \backslash PSL(2,{\mathbb Z})$. The Fourier and Poincar\'e 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 $w \leq 11$ the bound is saturated and we exhibit a basis for $\mathfrak{A}_w$.

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Source: https://tomesphere.com/paper/1902.04180