Notes on cluster algebras and some all-loop Feynman integrals

TL;DR

This paper investigates the cluster algebra structures underlying various all-loop Feynman integrals, revealing their symbol alphabets correspond to specific D-type cluster algebras and establishing connections to cluster configuration space.

Contribution

It identifies the cluster algebra types for several Feynman integrals and relates symbol letters to gauge-invariant cluster variables, extending understanding of their algebraic structure.

Findings

Penta-box ladder has D_3 alphabet, double-penta ladder likely D_4

Symbol letters relate to cluster configuration space u variables

Recursive predictions of higher-loop alphabets using Wilson-loop d log representation

Abstract

We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and (seven-point) double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $D_2\simeq A_1^2$, we show that penta-box ladder has an alphabet of $D_3\simeq A_3$ and provide strong evidence that the alphabet of double-penta ladder can be identified with a $D_4$ cluster algebra. We relate the symbol letters to the ${\bf u}$ variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop ${\rm d}\log$ representation, which allows us to predict higher-loop alphabet recursively; by applying such recursions to six-dimensional hexagon integrals, we also find $D_5$ and $D_6$ cluster functions for the two-mass-easy and three-mass-easy case, respectively.