Geometry from Integrability: Multi-Leg Fishnet Integrals in Two Dimensions
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arXiv
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| Format: | Preprint |
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2024
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| author | Duhr, Claude Klemm, Albrecht Loebbert, Florian Nega, Christoph Porkert, Franziska |
| author_facet | Duhr, Claude Klemm, Albrecht Loebbert, Florian Nega, Christoph Porkert, Franziska |
| contents | We generalise the geometric analysis of square fishnet integrals in two dimensions to the case of hexagonal fishnets with three-point vertices. Our results support the conjecture that fishnet Feynman integrals in two dimensions, together with their associated geometry, are completely fixed by their Yangian and permutation symmetries. As a new feature for the hexagonal fishnets, the star-triangle identity introduces an ambiguity in the graph representation of a given Feynman integral. This translates into a map between different geometric interpretations attached to a graph. We demonstrate explicitly how these fishnet integrals can be understood as Calabi-Yau varieties, whose Picard-Fuchs ideals are generated by the Yangian over the conformal algebra. In analogy to elliptic curves, which represent the simplest examples of fishnet integrals with four-point vertices, we find that the simplest examples of three-point fishnets correspond to Picard curves with natural generalisations at higher loop orders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_19034 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometry from Integrability: Multi-Leg Fishnet Integrals in Two Dimensions Duhr, Claude Klemm, Albrecht Loebbert, Florian Nega, Christoph Porkert, Franziska High Energy Physics - Theory We generalise the geometric analysis of square fishnet integrals in two dimensions to the case of hexagonal fishnets with three-point vertices. Our results support the conjecture that fishnet Feynman integrals in two dimensions, together with their associated geometry, are completely fixed by their Yangian and permutation symmetries. As a new feature for the hexagonal fishnets, the star-triangle identity introduces an ambiguity in the graph representation of a given Feynman integral. This translates into a map between different geometric interpretations attached to a graph. We demonstrate explicitly how these fishnet integrals can be understood as Calabi-Yau varieties, whose Picard-Fuchs ideals are generated by the Yangian over the conformal algebra. In analogy to elliptic curves, which represent the simplest examples of fishnet integrals with four-point vertices, we find that the simplest examples of three-point fishnets correspond to Picard curves with natural generalisations at higher loop orders. |
| title | Geometry from Integrability: Multi-Leg Fishnet Integrals in Two Dimensions |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2402.19034 |