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.
Highlights
Color-kinematics duality [6, 7], has had tremendous impact in the development of modern methods to compute scattering amplitudes: we refer the interested reader to the recent review [8] and references therein
Studying relations between amplitudes and constructing their bases amounts to characterizing these spaces. These spaces of integration cycles can be identified as homology groups with coefficients in a local system, often called twisted homology groups
The first question one should ask is, keeping all but one particle fixed, what are the bases of integration cycles in which higher-genus amplitudes are computed? This question can be addressed using twisted homology on a single Riemann surface with punctures, as opposed to the much richer twisted homology on the moduli space Mg,n, which is currently not understood
Summary
This section serves as an introduction to the framework used in the paper. It starts with a pedagogical example which illustrates how the language of local systems simplifies computations involving multi-valued functions. We review genus-zero string theory amplitudes in the context of twisted homologies and the interpretation of their monodromy relations in this language. We introduce the set-up which allow us to generalize the genus-zero computation to higher genera
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