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arXiv · 2402.19034

Geometry from Integrability: Multi-Leg Fishnet Integrals in Two Dimensions

Claude Duhr·Albrecht Klemm·Florian Loebbert·Christoph Nega·Franziska Porkert
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Abstract

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.

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Claude Duhr, Albrecht Klemm, Florian Loebbert, Christoph Nega, Franziska Porkert. 2024-02-29. Geometry from Integrability: Multi-Leg Fishnet Integrals in Two Dimensions. https://arxiv.org/abs/2402.19034

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