Holomorphic subgraph reduction of higher-point modular graph forms

TL;DR

This paper extends the holomorphic subgraph reduction technique from dihedral to trihedral modular graph forms, enabling simplification of complex functions in string theory amplitudes through a new regularization scheme.

Contribution

It introduces a modular covariant regularization scheme for conditionally convergent sums, facilitating the reduction of higher-point modular graph forms with arbitrary holomorphic subgraphs.

Findings

Extended reduction results to trihedral modular graph forms.

Developed a regularization scheme for conditionally convergent sums.

Enabled reduction of higher-point modular graph forms.

Abstract

Modular graph forms are a class of modular covariant functions which appear in the genus-one contribution to the low-energy expansion of closed string scattering amplitudes. Modular graph forms with holomorphic subgraphs enjoy the simplifying property that they may be reduced to sums of products of modular graph forms of strictly lower loop order. In the particular case of dihedral modular graph forms, a closed form expression for this holomorphic subgraph reduction was obtained previously by D'Hoker and Green. In the current work, we extend these results to trihedral modular graph forms. Doing so involves the identification of a modular covariant regularization scheme for certain conditionally convergent sums over discrete momenta, with some elements of the sum being excluded. The appropriate regularization scheme is identified for any number of exclusions, which in principle allows one to perform holomorphic subgraph reduction of higher-point modular graph forms with arbitrary holomorphic subgraphs.