Identities between Modular Graph Forms

TL;DR

The paper extends the study of modular graph functions by introducing modular graph forms, decorated graphs with independent holomorphic and anti-holomorphic weights, and by employing covariant derivatives and the Laplacian to map these forms within a unified framework. A central innovation is holomorphic subgraph reduction, which, combined with a sieve algorithm, relates non-holomorphic identities to holomorphic modular-form identities and enables systematic construction and proof of relations among modular graph forms. The authors apply this machinery to prove several conjectures from prior work (notably D4, D5, D3,1,1, and D2,2,1), establishing that certain polynomials in modular graph forms vanish identically at fixed weights, and thus revealing the internal algebraic structure of these objects. This approach not only provides short proofs for known relations but also offers a scalable, algorithmic path to discovering new identities at arbitrary weight, with potential connections to single-valued elliptic multiple polylogarithms and related number-theoretic structures. The work thus advances both the mathematical understanding of modular graph forms and their physical relevance in string theory amplitudes, while laying groundwork for automated discovery of modular identities.

Abstract

This paper investigates the relations between modular graph forms, which are generalizations of the modular graph functions that were introduced in earlier papers motivated by the structure of the low energy expansion of genus-one Type II superstring amplitudes. These modular graph forms are multiple sums associated with decorated Feynman graphs on the world-sheet torus. The action of standard differential operators on these modular graph forms admits an algebraic representation on the decorations. First order differential operators are used to map general non-holomorphic modular graph functions to holomorphic modular forms. This map is used to provide proofs of the identities between modular graph functions for weight less than six conjectured in earlier work, by mapping these identities to relations between holomorphic modular forms which are proven by holomorphic methods. The map is further used to exhibit the structure of identities at arbitrary weight.