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 , 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 toward decomposing divergent graphs. We provide a computer implementation for the manipulation of MGFs in the form of the Mathematica package ModularGraphForms which includes the basis decompositions obtained.

模图形式（MGFs）是一类非全纯模形式，自然出现在闭弦属一振幅的低能展开中，引起了纯数学家的极大兴趣。MGFs 满足许多非平凡的代数和微分关系，这些关系在文献中得到了广泛的研究，并导致了显着的简化。在本文中，我们系统地结合这些关系以获得总模权重的所有两点和三点 MGF 的基分解，从香蕉图的两个众所周知的身份开始。此外，我们研究了 MGF 积分表示中先前已知的关系，从而对作为 Kronecker-Eisenstein 级数的 Fay 恒等式的全纯子图减少有了新的理解，并为分解发散图打开了大门。我们以Mathematica包ModularGraphForms的形式提供了用于操作 MGF 的计算机实现，其中包括获得的基础分解。