Basis Decompositions and a Mathematica Package for Modular Graph Forms

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 $w+\bar{w}\le 12$, 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 $\text{Mathematica}$ package $\text{ModularGraphForms}$ which includes the basis decompositions obtained.

75+27 pages. Submission includes a Mathematica package and text files with basis decompositions in the ancillary files

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Cited by in corpus (9)

• The SAGEX Review on Scattering Amplitudes, Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their $\mathcal{N}=4$ supersymmetric Yang-Mills duals
• Modular graph forms from equivariant iterated Eisenstein integrals
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• To the cusp and back: Resurgent analysis for modular graph functions
• Integrating three-loop modular graph functions and transcendentality of string amplitudes
• Basis decompositions of genus-one string integrals
• Advanced tools for basis decompositions of genus-one string integrals
• Type II superstring amplitude at one-loop and transcendentality
• From Modular Graph Forms to Iterated Integrals