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