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

TL;DR

This work extends the geometry-integrability link for fishnet Feynman integrals from square to hexagonal 2D nets with three-point vertices, arguing that both square and hexagonal fishnets are fixed by Yangian and permutation symmetries. It develops a Calabi-Yau/motive framework where isotropic 2D fishnets correspond to CY or Picard geometries, with Picard-Fuchs ideals generated by the Yangian over the conformal algebra; star-triangle relations induce geometry-interpretation ambiguities, mapping to different CY/Picard realizations. The authors provide explicit low-loop examples showing that PFIs match the holomorphic Yangian differential ideals and that triangle-track graphs yield Picard curves/varieties whose periods realize the fishnet integrals. They emphasize that different geometric representations can describe the same motive, underscoring a motive-centric view for these integrals and offering analytic tools (hypergeometric and Appell-type functions) for exact evaluations. The results advance the program of deriving exact, symmetry-constrained expressions for multi-leg fishnet integrals and highlight rich mathematical structures connecting integrable quantum field theories, CY geometry, and algebraic geometry.

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.

Figures (20)

• Figure 1: The three regular tilings of the plane with vertices of valency $V=3,4,6$ respectively.
• Figure 2: Using the star-triangle relation in eq. (\ref{['eq:startriangle ']}), the triangular fishnet (red) can be transformed into a hexagonal fishnet (black) consisting of elementary three-point vertices only. There are two different possibilities for this identification (left and right figure), distinguished by the subset of vertices to which the star triangle-identity is applied.
• Figure 3: Illustration of the rule to associate evaluation parameters to a given Feynman graph. The evaluation parameters $s_j$ and $s_{j+1}$, which are associated to the external legs $j$ and $j+1$, respectively, are related by a term depending on the powers of external ($\nu$) and internal ($\mu$) propagators which lie on the propagator path that connects these external points.
• Figure 4: A three-loop fishnet integral before and after applying twice of the star-triangle relation. Black and red edges correspond to propagators raised to the powers $D/6$ and $D/3$, respectively.
• Figure 5: The two-loop graph $Z_2$ can be identified with a box graph (modulo an overall rational factor), when using the star-triangle identity. Black and red propagators are raised to powers $1/3$ and $2/3$, respectively.