论文标题
基础分解和模块化图表的Mathematica软件包
Basis Decompositions and a Mathematica Package for Modular Graph Forms
论文作者
论文摘要
模块图形式（MGF）是一类非旋晶模块化形式，它们自然出现在封闭弦属属的低能扩张中，并引起了纯数学家的极大兴趣。 MGF满足了许多非平凡的代数和差异关系，这些代数和差异关系已在文献中进行了广泛的研究并导致了重大的简化。在本文中，我们系统地结合了这些关系，以获取所有两点和三点MGF的基础分解，总模块化重量$ W+\ bar {w} \ leq12 $，仅从两个众所周知的香蕉图开始。此外，我们研究了MGF的整体表示中以前已知的关系，从而使人们对Holomorphic Subgraph降低的新理解是Kronecker的Fay身份 - Eisenstein系列，并为分解发散图打开了大门。我们为MGFS提供了计算机实现，以$ \ texttt {Mathematica} $ package $ \ texttt {modularGraphforms} $的形式，该{mathematica} $，其中包括获得的基本分解。
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.
