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Yangian-invariant fishnet integrals in 2 dimensions as volumes of Calabi-Yau varieties

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arxiv 2209.05291 v1 pith:RP4EMEBW submitted 2022-09-12 hep-th hep-phmath-phmath.MP

Claude Duhr , Albrecht Klemm , Florian Loebbert , Christoph Nega , Franziska Porkert This is my paper · ORCID
classification hep-thhep-phmath-phmath.MP
keywords calabi-yaudimensionsintegralintegralsmirrorvaluevolumeyangian-invariant
verification ladder T0 review T1 audit T2 compute T3 formal
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abstract

We argue that $\ell$-loop Yangian-invariant fishnet integrals in 2 dimensions are connected to a family of Calabi-Yau $\ell$-folds. The value of the integral can be computed from the periods of the Calabi-Yau, while the Yangian generators provide its Picard-Fuchs differential ideal. Using mirror symmetry, we can identify the value of the integral as the quantum volume of the mirror Calabi-Yau. We find that, similar to what happens in string theory, for $\ell=1$ and 2 the value of the integral agrees with the classical volume of the mirror, but starting from $\ell=3$, the classical volume gets corrected by instanton contributions. We illustrate these claims on several examples, and we use them to provide for the first time results for 2- and 3-loop Yangian-invariant traintrack integrals in 2 dimensions for arbitrary external kinematics.

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