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

TL;DR

The paper uncovers a deep link between Yangian-invariant 2D fishnet multi-loop integrals and families of Calabi-Yau $\ell$-folds, showing that integral values are governed by periods and Picard-Fuchs equations generated by the Yangian and graph automorphisms. Through mirror symmetry, the fishnet integrals are identified with the quantum volume of the mirror CY, reproducing classical volumes for $\ell\le2$ and incorporating instanton corrections for $\ell\ge3$. The authors provide explicit checks for 2- and 3-loop traintracks and 1-parameter rectangular fishnets, demonstrating a concrete computational bridge from Feynman integrals to geometric periods. This work inaugurates a volume interpretation of higher-loop Feynman integrals in 2D and opens pathways to new interactions between integrability, geometry, and perturbative QFT, with potential extensions to 4D and richer CY geometries.

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.

Figures (2)

• Figure 1: Ten-point five-loop fishnet integral cut out of a square tiling of the plane. If the $\ell = M\times N$ interior points span a rectangle, we denote the graph by $G_{M,N}$, with $M\le N$.
• Figure 2: The functions $\phi_{G_{1,2}}(z_1,z_2,z_3)$ and $\phi_{G_{1,3}}(z_1,\ldots,z_5)$ evaluated on the 1-dimensional slice $(z_1,z_2,z_3) = \tfrac{s}{16}(1,2,3)$ and $(z_1,\ldots,z_5) = \tfrac{s}{16}(1,2,12,4,5)$ (for the definition of our cross ratios, see the main text). The continuous lines represent the results obtained from our analytic result in terms of CY periods, while the dots are obtained from a numerical evaluation of the Feynman parameter representation of $G_{1,\ell}$.