Kinematics, cluster algebras and Feynman integrals

A bstract We identify cluster algebras for planar kinematics of conformal Feynman integrals in four dimensions, as sub-algebras of cluster algebras for Grassmannian G (4, n ) corresponding to n -point massless kinematics. We classify such algebras for cases through eight points, and provide evidence that singularities of the corresponding Feynman integrals are given by cluster variables and their algebraic generalizations. By sending a point to infinity, our results have implications for symbology of non-conformal Feynman integrals. We also find that dimensional reduction is achieved by folding and the resulting cluster algebras encode singularities of Feynman integrals in three dimensions. As a highly-nontrivial application, we study a eight-point three-loop wheel integral whose kinematics correspond to two-mass-easy box in the non-conformal limit: in addition to nine rational letters that are cluster variables, we also find three algebraic letters containing a new square root, in accordance with results from differential equations for two-loop case. Based on this alphabet, we bootstrap its symbol, which turns out to be strongly constrained by cluster adjacency conditions.