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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Acknowledgments
The work of FL was supported by funds of the Klaus Tschira Foundation gGmbH. This work was co-funded by the European Union (ERC Consolidator Grant LoCoMotive 101043686 (CD, FP) and ERC Starting Grant 949279 HighPHun (CN)). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. AK wants to thank Kilian Böhnisch and Matt Kerr for discussions. Special thanks to Matt Kerr for reading section 3.5 and suggesting improvements.
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Duhr, C., Klemm, A., Loebbert, F. et al. Geometry from integrability: multi-leg fishnet integrals in two dimensions. J. High Energ. Phys. 2024, 8 (2024). https://doi.org/10.1007/JHEP07(2024)008
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DOI: https://doi.org/10.1007/JHEP07(2024)008