Truncated cluster algebras and Feynman integrals with algebraic letters

A bstract We propose that the symbol alphabet for classes of planar, dual-conformal-invariant Feynman integrals can be obtained as truncated cluster algebras purely from their kinematics, which correspond to boundaries (compactifications of) G + (4 , n ) /T -particle massless kinematics. For one-, two-, three-mass-easy hexagon kinematics with = 7 8 9, we find finite D 4 5 and 6 respectively, in accordance previous result on alphabets these integrals. As main example, consider two massive corners opposite sides a affine algebra whose polytopal realization is co-dimension boundary 8) 39 facets; normal vectors 38 them g-vectors remaining one gives limit ray, yields an rational letters algebraic ones unique four-mass-box square root. construct space integrable symbols this physical first-entry conditions, dimension reduced using conditions version adjacency. Already at weight 4, by imposing last-entry inspired double-pentagon integral, are able uniquely determine part most generic integral. Finally, locate ladder up four loops differential equations derived Wilson-loop d log forms, remarkable pattern about appearance letters.