Positive geometries and canonical forms via mixed Hodge theory

TL;DR

This paper introduces a new mathematical framework using mixed Hodge theory to study positive geometries and their canonical forms, which are important in particle physics for understanding scattering amplitudes.

Contribution

It recasts positive geometries within mixed Hodge theory and identifies genus zero pairs as a natural setting, proving key properties of canonical forms.

Findings

Established basic properties of canonical forms within the new framework

Provided detailed examples including hyperplane arrangements and convex polytopes

Connected positive geometries to mixed Hodge theory for broader mathematical understanding

Abstract

''Positive geometries'' are a class of semi-algebraic domains which admit a unique ''canonical form'': a logarithmic form whose residues match the boundary structure of the domain. The study of such geometries is motivated by recent progress in particle physics, where the corresponding canonical forms are interpreted as the integrands of scattering amplitudes. We recast these concepts in the language of mixed Hodge theory, and identify ''genus zero pairs'' of complex algebraic varieties as a natural and general framework for the study of positive geometries and their canonical forms. In this framework, we prove some basic properties of canonical forms which have previously been proved or conjectured in the literature. We give many examples and study in detail the case of arrangements of hyperplanes and convex polytopes.