Monodromy relations from twisted homology

TL;DR

This paper reformulates monodromy relations in open-string scattering as boundary terms of twisted homologies, enabling explicit relations for any genus and external particles, and proposes a higher-genus generalization of the ! result.

Contribution

It introduces a novel formulation of monodromy relations using twisted homologies, applicable to arbitrary genus and particle number, and provides estimates on independent loop integrands.

Findings

Explicit linear relations for loop integrands at any genus.

Clarification of unphysical contributions in non-planar sectors.

Higher-genus ! generalization for independent integrands.

Abstract

We reformulate the monodromy relations of open-string scattering amplitudes as boundary terms of twisted homologies on the configuration spaces of Riemann surfaces of arbitrary genus. This allows us to write explicit linear relations involving loop integrands of open-string theories for any number of external particles and, for the first time, to arbitrary genus. In the non-planar sector, these relations contain seemingly unphysical contributions, which we argue clarify mismatches in previous literature. The text is mostly self-contained and presents a concise introduction to twisted homologies. As a result of this powerful formulation, we can propose estimates on the number of independent loop integrands based on Euler characteristics of the relevant configuration spaces, leading to a higher-genus generalization of the famous $(n-3)!$ result at genus zero.