Non-perturbative geometries for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes
Abstract There is a remarkable well-known connection between the G(4, n) cluster algebra and n-particle amplitudes in N $$ \mathcal{N} $$ = 4 SYM theory. For n ≥ 8 two long-standing open questions have been to find a mathematically natural way to identify a finite list of amplitude symbol letters fr...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-03-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP03(2021)065 |
| _version_ | 1818848097076248576 |
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| author | Nima Arkani-Hamed Thomas Lam Marcus Spradlin |
| author_facet | Nima Arkani-Hamed Thomas Lam Marcus Spradlin |
| author_sort | Nima Arkani-Hamed |
| collection | DOAJ |
| description | Abstract There is a remarkable well-known connection between the G(4, n) cluster algebra and n-particle amplitudes in N $$ \mathcal{N} $$ = 4 SYM theory. For n ≥ 8 two long-standing open questions have been to find a mathematically natural way to identify a finite list of amplitude symbol letters from among the infinitely many cluster variables, and to find an explanation for certain algebraic functions, such as the square roots of four-mass-box type, that are expected to appear in symbols but are not cluster variables. In this letter we use the notion of “stringy canonical forms” to construct polytopal realizations of certain compactifications of (the positive part of) the configuration space Conf n (ℙ k−1) ≅ G(k, n)/T that are manifestly finite for all k and n. Some facets of these polytopes are naturally associated to cluster variables, while others are naturally associated to algebraic functions constructed from Lusztig’s canonical basis. For (k, n) = (4, 8) the latter include precisely the expected square roots, revealing them to be related to certain “overpositive” functions of the kinematical invariants. |
| first_indexed | 2024-12-19T06:11:55Z |
| format | Article |
| id | doaj.art-d94e0d2e8ac14fcfbc596ca292b3d097 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-19T06:11:55Z |
| publishDate | 2021-03-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-d94e0d2e8ac14fcfbc596ca292b3d0972022-12-21T20:32:59ZengSpringerOpenJournal of High Energy Physics1029-84792021-03-012021311510.1007/JHEP03(2021)065Non-perturbative geometries for planar N $$ \mathcal{N} $$ = 4 SYM amplitudesNima Arkani-Hamed0Thomas Lam1Marcus Spradlin2School of Natural Sciences, Institute for Advanced StudyDepartment of Mathematics, University of MichiganDepartment of Physics and Brown Theoretical Physics Center, Brown UniversityAbstract There is a remarkable well-known connection between the G(4, n) cluster algebra and n-particle amplitudes in N $$ \mathcal{N} $$ = 4 SYM theory. For n ≥ 8 two long-standing open questions have been to find a mathematically natural way to identify a finite list of amplitude symbol letters from among the infinitely many cluster variables, and to find an explanation for certain algebraic functions, such as the square roots of four-mass-box type, that are expected to appear in symbols but are not cluster variables. In this letter we use the notion of “stringy canonical forms” to construct polytopal realizations of certain compactifications of (the positive part of) the configuration space Conf n (ℙ k−1) ≅ G(k, n)/T that are manifestly finite for all k and n. Some facets of these polytopes are naturally associated to cluster variables, while others are naturally associated to algebraic functions constructed from Lusztig’s canonical basis. For (k, n) = (4, 8) the latter include precisely the expected square roots, revealing them to be related to certain “overpositive” functions of the kinematical invariants.https://doi.org/10.1007/JHEP03(2021)065Scattering AmplitudesSupersymmetric Gauge Theory |
| spellingShingle | Nima Arkani-Hamed Thomas Lam Marcus Spradlin Non-perturbative geometries for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes Journal of High Energy Physics Scattering Amplitudes Supersymmetric Gauge Theory |
| title | Non-perturbative geometries for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes |
| title_full | Non-perturbative geometries for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes |
| title_fullStr | Non-perturbative geometries for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes |
| title_full_unstemmed | Non-perturbative geometries for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes |
| title_short | Non-perturbative geometries for planar N $$ \mathcal{N} $$ = 4 SYM amplitudes |
| title_sort | non perturbative geometries for planar n mathcal n 4 sym amplitudes |
| topic | Scattering Amplitudes Supersymmetric Gauge Theory |
| url | https://doi.org/10.1007/JHEP03(2021)065 |
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