Communications in Number Theory and Physics

Volume 16 (2022)

Number 1

Identities among higher genus modular graph tensors

Pages: 35 – 74



Eric D’Hoker (Institute for Theoretical Physics, Department of Physics and Astronomy, University of California, Los Angeles, Calif., U.S.A.)

Oliver Schlotterer (Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden)


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.


higher-genus modular form, string scattering amplitude

The research of Eric D’Hoker was supported in part by NSF grant PHY-19-14412.

The research of Oliver Schlotterer was supported by the European Research Council under ERCSTG-804286 UNISCAMP.

Received 22 December 2020

Accepted 14 September 2021

Published 1 February 2022