  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.
