Relations between elliptic modular graphs

TL;DR

This paper studies elliptic modular graph functions from genus two string invariants, showing they can be simplified to graphs with only Green function links, reducing the basis complexity.

Contribution

It introduces a method to express elliptic modular graphs with derivatives as graphs with only Green functions, simplifying their structure.

Findings

Graphs with derivatives can be expressed in terms of Green function-only graphs.

Reduces the basis size of elliptic modular graphs.

Simplifies the analysis of genus two string invariants.

Abstract

We consider certain elliptic modular graph functions that arise in the asymptotic expansion around the non--separating node of genus two string invariants that appear in the integrand of the $D^8 R^4$ interaction in the low momentum expansion of the four graviton amplitude in type II superstring theory. These elliptic modular graphs have links given by the Green function, as well its holomorphic and anti--holomorphic derivatives. Using appropriate auxiliary graphs at various intermediate stages of the analysis, we show that each graph can be expressed solely in terms of graphs with links given only by the Green function and not its derivatives. This results in a reduction in the number of basis elements in the space of elliptic modular graphs.