Symbolic integration and multiple polylogarithms

TL;DR

The paper addresses symbolic evaluation of Feynman integrals by framing them in a robust algebraic setting of iterated integrals tied to multiple polylogarithms. It develops a universal, multivariable polylogarithm framework using Chen's homotopy invariance and a map that relates one-variable to multi-variable cases, organized via the bar-construction of integrable words. Key contributions include the explicit construction of the integrable word space $\mathcal{B}(\Omega_n)$, an explicit basis obtained through the $\Psi$ map, and a practical pathway to express high-dimensional integrals as $\mathcal{Z}$-linear combinations of lower-dimensional bases, enabling algorithmic computation. The work paves the way for automated, symbolic evaluation of a broad class of Feynman parameter integrals and suggests extensions to more general graphs and gauge theories.

Abstract

We review a method for the algebraic treatment of a family of functions which contains the multiple polylogarithms, with applications to the symbolic calculation of Feynman integrals.