Positive geometries and canonical forms via mixed Hodge theory
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911145231646720 |
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| author | Brown, Francis Dupont, Clément |
| author_facet | Brown, Francis Dupont, Clément |
| contents | ''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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03202 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Positive geometries and canonical forms via mixed Hodge theory Brown, Francis Dupont, Clément Algebraic Geometry High Energy Physics - Theory Mathematical Physics ''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. |
| title | Positive geometries and canonical forms via mixed Hodge theory |
| topic | Algebraic Geometry High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2501.03202 |