Stringy canonical forms
Abstract Canonical forms of positive geometries play an important role in revealing hidden structures of scattering amplitudes, from amplituhedra to associahedra. In this paper, we introduce “stringy canonical forms”, which provide a natural definition and extension of canonical for...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Springer Berlin Heidelberg
2021
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| Online Access: | https://hdl.handle.net/1721.1/132098 |
| _version_ | 1811077881647857664 |
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| author | Arkani-Hamed, Nima He, Song Lam, Thomas |
| author_facet | Arkani-Hamed, Nima He, Song Lam, Thomas |
| author_sort | Arkani-Hamed, Nima |
| collection | MIT |
| description | Abstract
Canonical forms of positive geometries play an important role in revealing hidden structures of scattering amplitudes, from amplituhedra to associahedra. In this paper, we introduce “stringy canonical forms”, which provide a natural definition and extension of canonical forms for general polytopes, deformed by a parameter α′. They are defined by real or complex integrals regulated with polynomials with exponents, and are meromorphic functions of the exponents, sharing various properties of string amplitudes. As α′→ 0, they reduce to the usual canonical form of a polytope given by the Minkowski sum of the Newton polytopes of the regulating polynomials, or equivalently the volume of the dual of this polytope, naturally determined by tropical functions. At finite α′, they have simple poles corresponding to the facets of the polytope, with the residue on the pole given by the stringy canonical form of the facet. There is the remarkable connection between the α′→ 0 limit of tree-level string amplitudes, and scattering equations that appear when studying the α′→ ∞ limit. We show that there is a simple conceptual understanding of this phenomenon for any stringy canonical form: the saddle-point equations provide a diffeomorphism from the integration domain to the interior of the polytope, and thus the canonical form can be obtained as a pushforward via summing over saddle points. When the stringy canonical form is applied to the ABHY associahedron in kinematic space, it produces the usual Koba-Nielsen string integral, giving a direct path from particle to string amplitudes without an a priori reference to the string worldsheet. We also discuss a number of other examples, including stringy canonical forms for finite-type cluster algebras (with type A corresponding to usual string amplitudes), and other natural integrals over the positive Grassmannian. |
| first_indexed | 2024-09-23T10:49:46Z |
| format | Article |
| id | mit-1721.1/132098 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T10:49:46Z |
| publishDate | 2021 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1320982021-09-21T03:47:12Z Stringy canonical forms Arkani-Hamed, Nima He, Song Lam, Thomas Abstract Canonical forms of positive geometries play an important role in revealing hidden structures of scattering amplitudes, from amplituhedra to associahedra. In this paper, we introduce “stringy canonical forms”, which provide a natural definition and extension of canonical forms for general polytopes, deformed by a parameter α′. They are defined by real or complex integrals regulated with polynomials with exponents, and are meromorphic functions of the exponents, sharing various properties of string amplitudes. As α′→ 0, they reduce to the usual canonical form of a polytope given by the Minkowski sum of the Newton polytopes of the regulating polynomials, or equivalently the volume of the dual of this polytope, naturally determined by tropical functions. At finite α′, they have simple poles corresponding to the facets of the polytope, with the residue on the pole given by the stringy canonical form of the facet. There is the remarkable connection between the α′→ 0 limit of tree-level string amplitudes, and scattering equations that appear when studying the α′→ ∞ limit. We show that there is a simple conceptual understanding of this phenomenon for any stringy canonical form: the saddle-point equations provide a diffeomorphism from the integration domain to the interior of the polytope, and thus the canonical form can be obtained as a pushforward via summing over saddle points. When the stringy canonical form is applied to the ABHY associahedron in kinematic space, it produces the usual Koba-Nielsen string integral, giving a direct path from particle to string amplitudes without an a priori reference to the string worldsheet. We also discuss a number of other examples, including stringy canonical forms for finite-type cluster algebras (with type A corresponding to usual string amplitudes), and other natural integrals over the positive Grassmannian. 2021-09-20T17:41:57Z 2021-09-20T17:41:57Z 2021-02-09 2021-04-04T03:52:38Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/132098 Journal of High Energy Physics. 2021 Feb 09;2021(2):69 PUBLISHER_CC en https://doi.org/10.1007/JHEP02(2021)069 Creative Commons Attribution https://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Arkani-Hamed, Nima He, Song Lam, Thomas Stringy canonical forms |
| title | Stringy canonical forms |
| title_full | Stringy canonical forms |
| title_fullStr | Stringy canonical forms |
| title_full_unstemmed | Stringy canonical forms |
| title_short | Stringy canonical forms |
| title_sort | stringy canonical forms |
| url | https://hdl.handle.net/1721.1/132098 |
| work_keys_str_mv | AT arkanihamednima stringycanonicalforms AT hesong stringycanonicalforms AT lamthomas stringycanonicalforms |