Proving relations between modular graph functions

Understand

We consider modular graph functions that arise in the low energy expansion of the four graviton amplitude in type II string theory.

• The vertices of these graphs are the positions of insertions of vertex operators on the toroidal worldsheet, while the links are the scalar Green functions connecting the vertices.
• Graphs with four and five links satisfy several non--trivial relations, which have been proved recently.
• We prove these relations by using elementary properties of Green functions and the details of the graphs.