Positive Genus Pairs from Amplituhedra
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910024333262848 |
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| author | Koefler, Joris Pavlov, Dmitrii Sinn, Rainer |
| author_facet | Koefler, Joris Pavlov, Dmitrii Sinn, Rainer |
| contents | A main conjecture in the field of Positive Geometry states that amplituhedra, which are certain semi-algebraic sets in the Grassmannian, are positive geometries. It is motivated by examples showing that the canonical forms of certain amplituhedra compute scattering amplitudes in particle physics. Beyond a small number of special cases, this conjecture is still open. In recent work, Brown and Dupont introduced a new framework, based on mixed Hodge theory, connecting canonical forms and de Rham cohomology via genus zero pairs. We give short proofs that the amplituhedron gives rise to a genus zero pair in the cases when it is known to be a positive geometry. However, in the general case we show that amplituhedra inside the Grassmannian give rise to pairs of strictly positive genus. We provide an explicit example of a genus one pair arising from a positive geometry in projective space, showing that having genus zero is not a necessary condition to be a positive geometry. Finally, we show that this positive geometry still gives rise to a genus zero pair in a different ambient variety. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11142 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Positive Genus Pairs from Amplituhedra Koefler, Joris Pavlov, Dmitrii Sinn, Rainer Algebraic Geometry High Energy Physics - Theory Mathematical Physics A main conjecture in the field of Positive Geometry states that amplituhedra, which are certain semi-algebraic sets in the Grassmannian, are positive geometries. It is motivated by examples showing that the canonical forms of certain amplituhedra compute scattering amplitudes in particle physics. Beyond a small number of special cases, this conjecture is still open. In recent work, Brown and Dupont introduced a new framework, based on mixed Hodge theory, connecting canonical forms and de Rham cohomology via genus zero pairs. We give short proofs that the amplituhedron gives rise to a genus zero pair in the cases when it is known to be a positive geometry. However, in the general case we show that amplituhedra inside the Grassmannian give rise to pairs of strictly positive genus. We provide an explicit example of a genus one pair arising from a positive geometry in projective space, showing that having genus zero is not a necessary condition to be a positive geometry. Finally, we show that this positive geometry still gives rise to a genus zero pair in a different ambient variety. |
| title | Positive Genus Pairs from Amplituhedra |
| topic | Algebraic Geometry High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2601.11142 |