Relations between elliptic modular graphs

Anirban Basu

doi:10.1007/jhep12(2020)195

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 D8ℛ4 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.

Highlights

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 D8R4 interaction in the low momentum expansion of the four graviton amplitude in type II superstring theory
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
Consider the string invariants that arise in the evaluation of the D8R4 and D6R5 interactions in the low momentum expansion of the four and five graviton amplitudes respectively, at genus two

Summary

Relations between the various modular graphs

It is evaluated trivially using (2.4) for the link having a ∂ as well as a ∂ acting on the Green function, and alternatively by moving the derivatives around the circuit and using (2.4). Using the definitions in (A.1), and the relations (A.4) and (A.12) we see that the we have expressed H3(v) in terms of graphs without derivatives of the Green function as their links. From the various relations above, we see that K1(v) is expressible in terms of the modular graphs given in figures 3 and 4, in which the links are given by the Green function and not its derivatives. We have obtained non-trivial relations involving the elliptic modular graph functions given by (3.19), (3.28), (3.39) and (3.47).

Some consistency checks

A Some elliptic modular graphs and useful identities