Tropical Geometry, Quantum Affine Algebras, and Scattering Amplitudes
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929312771342336 |
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| author | Early, Nick Li, Jian-Rong |
| author_facet | Early, Nick Li, Jian-Rong |
| contents | The goal of this paper is to make a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics. The connection allows us to study important and difficult questions in these areas:
(1) We give a systematic construction of prime modules (including prime non-real modules) of quantum affine algebras using tropical geometry. We also introduce new objects which generalize positive tropical Grassmannians.
(2) We propose a generalization of Grassmannian string integrals in physics, in which the integrand is no longer a finite, but rather an infinite product indexed by prime modules of a quantum affine algebra. We give a general formula of $u$-variables using prime tableaux (corresponding to prime modules of quantum affine algebras of type $A$) and Auslander-Reiten quivers of Grassmannian cluster categories.
(3) We study limit $g$-vectors of cluster algebras. This is another way to obtain prime non-real modules of quantum affine algebras systematically. Using limit $g$-vectors, we construct new examples of non-real modules of quantum affine algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_05618 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Tropical Geometry, Quantum Affine Algebras, and Scattering Amplitudes Early, Nick Li, Jian-Rong Quantum Algebra High Energy Physics - Theory Combinatorics Representation Theory The goal of this paper is to make a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics. The connection allows us to study important and difficult questions in these areas: (1) We give a systematic construction of prime modules (including prime non-real modules) of quantum affine algebras using tropical geometry. We also introduce new objects which generalize positive tropical Grassmannians. (2) We propose a generalization of Grassmannian string integrals in physics, in which the integrand is no longer a finite, but rather an infinite product indexed by prime modules of a quantum affine algebra. We give a general formula of $u$-variables using prime tableaux (corresponding to prime modules of quantum affine algebras of type $A$) and Auslander-Reiten quivers of Grassmannian cluster categories. (3) We study limit $g$-vectors of cluster algebras. This is another way to obtain prime non-real modules of quantum affine algebras systematically. Using limit $g$-vectors, we construct new examples of non-real modules of quantum affine algebras. |
| title | Tropical Geometry, Quantum Affine Algebras, and Scattering Amplitudes |
| topic | Quantum Algebra High Energy Physics - Theory Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2303.05618 |