Cohomologies of Configuration Spaces and Higher-Dimensional Polylogarithms in Renormalization Group Problems

The deviation from commutativity of the renormalization and the action of all linear partial differential operators is the main source of the anomalies in quantum field theory, including the renormalization group action. This deviation is characterized by certain 'renormalization cocycles' that are related to cohomologies of the so called (ordered) configuration spaces. Cohomological differential equations that determine the renormalization cocycles up to the renormalization freedom are obtained. The solution of these equations requires introducing transcendental extensions related to higher-dimensional polylogarithms.