Algebraic identities between families of (elliptic) modular graphs

TL;DR

This paper develops a method to generate infinite algebraic identities between elliptic modular graphs by convoluting seed identities with families of graphs, involving unintegrated vertices and integration over the worldsheet.

Contribution

It introduces a systematic process to produce new identities between elliptic modular graphs through convolution with arbitrary graph expressions, expanding the known algebraic relations.

Findings

Generated infinite new identities between elliptic modular graphs.

Demonstrated convolution with seed identities produces parametrized algebraic relations.

Identified identities involving all vertices integrated over the worldsheet.

Abstract

Consider an algebraic identity between elliptic modular graphs where several vertices are at fixed locations (and hence unintegrated) while the others are integrated over the toroidal worldsheet. At any unintegrated vertex, we can glue an arbitrary expression involving elliptic modular graphs which has the same unintegrated vertex. Integrating over that vertex, we obtain new algebraic identities between elliptic modular graphs. Hence this elementary process of convoluting the original "seed" identity with other graphs yields infinite number of new identities. We consider various seed identities in which two of the vertices are unintegrated. Convoluting them with families of elliptic modular graphs, we obtain new identities. Each identity is parametrized by an arbitrary number of links in the graphs as well as the positions of unintegrated vertices. On identifying the unintegrated vertices, this leads to an algebraic identity involving modular graphs where all the vertices are integrated over the worldsheet.