Reductions of GKZ Systems and Applications to Cosmological Correlators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910887602814976 |
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| author | Grimm, Thomas W. Hoefnagels, Arno |
| author_facet | Grimm, Thomas W. Hoefnagels, Arno |
| contents | A powerful approach to computing Feynman integrals or cosmological correlators is to consider them as solution to systems of differential equations. Often these can be chosen to be Gelfand-Kapranov-Zelevinsky (GKZ) systems. However, their naive construction introduces a significant amount of unnecessary complexity. In this paper we present an algorithm which allows for reducing these GKZ systems to smaller subsystems if a parameter associated to the GKZ systems is resonant. These simpler subsystems can then be solved separately resulting in solutions for the full system. The algorithm makes it possible to check when reductions happen and allows for finding the associated simpler solutions. While originating in the mathematical theory of D-modules analyzed via exact sequences of Euler-Koszul homologies, the algorithm can be used without knowledge of this framework. We motivate the need for such reduction techniques by considering cosmological correlators on an FRW space-time and solve the tree-level single-exchange correlator in this way. It turns out that this integral exemplifies an interesting relation between locality and the reduction of the differential equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13815 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reductions of GKZ Systems and Applications to Cosmological Correlators Grimm, Thomas W. Hoefnagels, Arno High Energy Physics - Theory General Relativity and Quantum Cosmology Algebraic Geometry A powerful approach to computing Feynman integrals or cosmological correlators is to consider them as solution to systems of differential equations. Often these can be chosen to be Gelfand-Kapranov-Zelevinsky (GKZ) systems. However, their naive construction introduces a significant amount of unnecessary complexity. In this paper we present an algorithm which allows for reducing these GKZ systems to smaller subsystems if a parameter associated to the GKZ systems is resonant. These simpler subsystems can then be solved separately resulting in solutions for the full system. The algorithm makes it possible to check when reductions happen and allows for finding the associated simpler solutions. While originating in the mathematical theory of D-modules analyzed via exact sequences of Euler-Koszul homologies, the algorithm can be used without knowledge of this framework. We motivate the need for such reduction techniques by considering cosmological correlators on an FRW space-time and solve the tree-level single-exchange correlator in this way. It turns out that this integral exemplifies an interesting relation between locality and the reduction of the differential equations. |
| title | Reductions of GKZ Systems and Applications to Cosmological Correlators |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology Algebraic Geometry |
| url | https://arxiv.org/abs/2409.13815 |