Abstract
We analyse the family of Calabi-Yau varieties attached to four-point fishnet integrals in two dimensions. We find that the Picard-Fuchs operators for fishnet integrals are exterior powers of the Picard-Fuchs operators for ladder integrals. This implies that the periods of the Calabi-Yau varieties for fishnet integrals can be written as determinants of periods for ladder integrals. The representation theory of the geometric monodromy group plays an important role in this context. We then show how the determinant form of the periods immediately leads to the well-known Basso-Dixon formula for four-point fishnet integrals in two dimensions. Notably, the relation to Calabi-Yau geometry implies that the volume is also expressible via a determinant formula of Basso-Dixon type. Finally, we show how the fishnet integrals can be written in terms of iterated integrals naturally attached to the Calabi-Yau varieties.
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Acknowledgments
The authors thank V. Kazakov. We want, in particular, to thank Matt Kerr for his visit to the MPIM in Bonn and the discussions on representation theoretic questions in Hodge Theory. CN was supported by the Excellence Cluster ORIGINS funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2094 - 390783311. The work of FL is 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.
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Duhr, C., Klemm, A., Loebbert, F. et al. The Basso-Dixon formula and Calabi-Yau geometry. J. High Energ. Phys. 2024, 177 (2024). https://doi.org/10.1007/JHEP03(2024)177
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DOI: https://doi.org/10.1007/JHEP03(2024)177