Basis Decompositions and a Mathematica Package for Modular Graph Forms

TL;DR

This paper develops basis decompositions for modular graph forms relevant in string theory, introduces new algebraic relations, and provides a Mathematica package for their manipulation, advancing both mathematical understanding and computational tools.

Contribution

It systematically derives basis decompositions for two- and three-point MGFs up to weight 12 and introduces a Mathematica package for their analysis.

Findings

Basis decompositions for MGFs up to weight 12

New understanding of holomorphic subgraph reduction as Fay identities

A Mathematica package for manipulating MGFs

Abstract

Modular graph forms (MGFs) are a class of non-holomorphic modular forms which naturally appear in the low-energy expansion of closed-string genus-one amplitudes and have generated considerable interest from pure mathematicians. MGFs satisfy numerous non-trivial algebraic- and differential relations which have been studied extensively in the literature and lead to significant simplifications. In this paper, we systematically combine these relations to obtain basis decompositions of all two- and three-point MGFs of total modular weight $w+\bar{w}\leq12$, starting from just two well-known identities for banana graphs. Furthermore, we study previously known relations in the integral representation of MGFs, leading to a new understanding of holomorphic subgraph reduction as Fay identities of Kronecker--Eisenstein series and opening the door towards decomposing divergent graphs. We provide a computer implementation for the manipulation of MGFs in the form of the $\texttt{Mathematica}$ package $\texttt{ModularGraphForms}$ which includes the basis decompositions obtained.