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

TL;DR

This paper extends the geometric analysis of fishnet Feynman integrals in two dimensions to hexagonal configurations, revealing their connection to Calabi-Yau varieties and Picard curves, and emphasizing the role of symmetries and identities in their structure.

Contribution

It introduces a geometric framework for hexagonal fishnet integrals, linking them to Calabi-Yau varieties and exploring the implications of star-triangle identities on their representations.

Findings

Fishnet integrals can be understood as Calabi-Yau varieties.

Star-triangle identity introduces ambiguity in graph representations.

Three-point fishnets relate to Picard curves, generalizing elliptic curve cases.

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.