Identities among higher genus 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.

arXiv comment37 pages, v2: added a few figures and explanations; version to appear in Communications in Number Theory and Physics

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• Integrating simple genus two string invariants over moduli space