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 , 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.
Highlights
Scattering amplitudes in string theory have in recent years experienced a rise in interest due to the rich mathematical structures appearing in their calculation and their close relations to field-theory amplitudes
We systematically study relations between modular graph forms, a class of functions of the modular parameter τ = τ1 + iτ2, τ1, τ2 ∈ R with τ2 > 0 which make the computation of the low-energy expansion of the integrals over the punctures algorithmic and have been studied widely in the literature [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]
This paper is structured as follows: In Section 2, we review the definition of modular graph forms and their different representations as well as some other important objects
Summary
Scattering amplitudes in string theory have in recent years experienced a rise in interest due to the rich mathematical structures appearing in their calculation and their close relations to field-theory amplitudes. We systematically study relations between modular graph forms (mgfs), a class of functions of the modular parameter τ = τ1 + iτ, τ1, τ2 ∈ R with τ2 > 0 which make the computation of the low-energy expansion of the integrals over the punctures algorithmic and have been studied widely in the literature [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. Mgfs satisfy many non-trivial differential equations, we will focus here mainly on algebraic relations
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