Notes on cluster algebras and some all-loop Feynman integrals

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 has an of $D_3\simeq A_3$ provide strong evidence double-penta can be identified with a $D_4$ algebra. relate letters ${\bf u}$ variables configuration space, which gauge-invariant description algebra, find various sub-algebras associated limits comment on constraints similar extended-Steinmann relations or adjacency conditions function spaces. Our based recently proposed Wilson-loop ${\rm d}\log$ representation, allows us predict higher-loop recursively; by applying such recursions six-dimensional hexagon also $D_5$ $D_6$ functions two-mass-easy three-mass-easy case, respectively.