Cluster Algebras for Feynman Integrals

We initiate the study of cluster algebras in Feynman integrals dimensional regularization. provide evidence that four-point with one off-shell leg are described by a $C_{2}$ algebra, and we find adjacency relations restrict allowed function space. By embedding inside $A_3$ identify these adjacencies extended Steinmann for six-particle massless scattering. The algebra connection restricts functions space vector boson or Higgs plus jet amplitudes, form factors recently considered $\mathcal{N}=4$ super Yang-Mills. explain general procedures studying relationships between alphabets generalized polylogarithmic algebras, use them to various identifications one-loop algebras. In particular, show how can obtain five-particle scattering from discussed dual conformal eight-particle alphabet related $G(4,8)$ algebra.