Minimal Kinematics: An All and Peek into Trop+G( , )
In this note, we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM) generalized biadjoint amplitudes for all and n on what we call the minimal kinematics. We prove that on the minimal kinematics, the scattering equations on the configuration space of points on ℂℙᵏ⁻¹ have a unique solu...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2021 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2021
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211345 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Minimal Kinematics: An All and Peek into Trop+G( , ). Freddy Cachazo and Nick Early. SIGMA 17 (2021), 078, 22 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859746222156283904 |
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| author | Cachazo, Freddy Early, Nick |
| author_facet | Cachazo, Freddy Early, Nick |
| citation_txt | Minimal Kinematics: An All and Peek into Trop+G( , ). Freddy Cachazo and Nick Early. SIGMA 17 (2021), 078, 22 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this note, we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM) generalized biadjoint amplitudes for all and n on what we call the minimal kinematics. We prove that on the minimal kinematics, the scattering equations on the configuration space of points on ℂℙᵏ⁻¹ have a unique solution, and that this solution is in the image of a Veronese embedding. The minimal kinematics is a generalization of the one recently introduced by Early for = 2 and uses a choice of cyclic ordering. We conjecture an explicit formula for ⁽ᵏ⁾ₙ( , ), which we have checked analytically through = 10 for all k. The answer is a simple rational function which has only simple poles; the poles have the combinatorial structure of the circulant graph C⁽¹’²’ ˙˙˙’ ᵏ⁻²⁾ₙ. Generalized biadjoint amplitudes can also be evaluated using the positive tropical Grassmannian Tr+G( , ) in terms of generalized planar Feynman diagrams. We find perfect agreement between both definitions for all cases where the latter is known in the literature. In particular, this gives the first strong consistency check on the 90608 planar arrays for Tr+G(4,8) recently computed by Cachazo, Guevara, Umbert, and Zhang. We also introduce another class of special kinematics called planar-basis kinematics, which generalizes the one introduced by Cachazo, He, and Yuan for = 2 and uses the planar basis recently introduced by Early for all . Based on numerical computations through = 8 for all k, we conjecture that on the planar-basis kinematics ⁽ᵏ⁾ₙ( , ) evaluates to the multidimensional Catalan numbers, suggesting the possibility of novel combinatorial interpretations. For = 2, these are the standard Catalan numbers.
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| first_indexed | 2026-03-15T16:30:07Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211345 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T16:30:07Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
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| spelling | Cachazo, Freddy Early, Nick 2025-12-30T15:52:32Z 2021 Minimal Kinematics: An All and Peek into Trop+G( , ). Freddy Cachazo and Nick Early. SIGMA 17 (2021), 078, 22 pages 1815-0659 2020 Mathematics Subject Classification: 14M15; 05E99; 14T99 arXiv:2003.07958 https://nasplib.isofts.kiev.ua/handle/123456789/211345 https://doi.org/10.3842/SIGMA.2021.078 In this note, we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM) generalized biadjoint amplitudes for all and n on what we call the minimal kinematics. We prove that on the minimal kinematics, the scattering equations on the configuration space of points on ℂℙᵏ⁻¹ have a unique solution, and that this solution is in the image of a Veronese embedding. The minimal kinematics is a generalization of the one recently introduced by Early for = 2 and uses a choice of cyclic ordering. We conjecture an explicit formula for ⁽ᵏ⁾ₙ( , ), which we have checked analytically through = 10 for all k. The answer is a simple rational function which has only simple poles; the poles have the combinatorial structure of the circulant graph C⁽¹’²’ ˙˙˙’ ᵏ⁻²⁾ₙ. Generalized biadjoint amplitudes can also be evaluated using the positive tropical Grassmannian Tr+G( , ) in terms of generalized planar Feynman diagrams. We find perfect agreement between both definitions for all cases where the latter is known in the literature. In particular, this gives the first strong consistency check on the 90608 planar arrays for Tr+G(4,8) recently computed by Cachazo, Guevara, Umbert, and Zhang. We also introduce another class of special kinematics called planar-basis kinematics, which generalizes the one introduced by Cachazo, He, and Yuan for = 2 and uses the planar basis recently introduced by Early for all . Based on numerical computations through = 8 for all k, we conjecture that on the planar-basis kinematics ⁽ᵏ⁾ₙ( , ) evaluates to the multidimensional Catalan numbers, suggesting the possibility of novel combinatorial interpretations. For = 2, these are the standard Catalan numbers. We would like to thank A. Guevara for useful discussions. This research was supported in part by a grant from the Gluskin Sheff/Onex Freeman Dyson Chair in Theoretical Physics and by Perimeter Institute. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Minimal Kinematics: An All and Peek into Trop+G( , ) Article published earlier |
| spellingShingle | Minimal Kinematics: An All and Peek into Trop+G( , ) Cachazo, Freddy Early, Nick |
| title | Minimal Kinematics: An All and Peek into Trop+G( , ) |
| title_full | Minimal Kinematics: An All and Peek into Trop+G( , ) |
| title_fullStr | Minimal Kinematics: An All and Peek into Trop+G( , ) |
| title_full_unstemmed | Minimal Kinematics: An All and Peek into Trop+G( , ) |
| title_short | Minimal Kinematics: An All and Peek into Trop+G( , ) |
| title_sort | minimal kinematics: an all and peek into trop+g( , ) |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211345 |
| work_keys_str_mv | AT cachazofreddy minimalkinematicsanallandpeekintotropg AT earlynick minimalkinematicsanallandpeekintotropg |