Identities among higher genus modular graph tensors

TL;DR

This paper establishes algebraic identities among higher genus modular graph tensors, extending Kawazumi's work, and applies to various loop orders, revealing deep symmetries in the mathematical structure of Feynman graphs on Riemann surfaces.

Contribution

It introduces new algebraic identities for higher genus modular graph tensors, generalizing previous results and covering arbitrary genus, tensor rank, and loop order.

Findings

Proves identities between one-loop and tree-level tensors for all genera.

Derives identities applicable to higher loop order tensors.

Extends Kawazumi's construction to a broader class of modular graph tensors.

Abstract

Higher genus modular graph tensors map Feynman graphs to functions on the Torelli space of genus-$h$ compact Riemann surfaces which transform as tensors under the modular group $Sp(2h , \mathbb Z)$, thereby generalizing a construction of Kawazumi. An infinite family of algebraic identities between one-loop and tree-level modular graph tensors are proven for arbitrary genus and arbitrary tensorial rank. We also derive a family of identities that apply to modular graph tensors of higher loop order.