Polylogarithm Identities, Cluster Algebras and the$$\mathcal {N} = 4$$ Supersymmetric Theory

Abstract

Scattering amplitudes in super-Yang Mills theory can be computed to higher perturbative orders than in any other four-dimensional quantum field theory. The results are interesting transcendental functions. By a hidden symmetry (dual conformal symmetry) the arguments of these functions have a geometric interpretation in terms of configurations of points in and they turn out to be cluster coordinates. We briefly introduce cluster algebras and discuss their Poisson structure and the Sklyanin bracket. Finally, we present a-term trilogarithm identity which was discovered by accident while studying the physical results.