Feynman Graphs

TL;DR

This paper reviews Feynman graphs and integrals, focusing on multiple polylogarithms and algebraic geometry methods for complex integrals beyond polylogarithmic expressions.

Contribution

It provides a comprehensive overview of the algebraic properties of multiple polylogarithms and discusses advanced techniques for evaluating complex Feynman integrals.

Findings

Multiple polylogarithms are central to Feynman integral evaluation.

Algebraic geometry offers tools for integrals beyond polylogarithmic functions.

The paper summarizes key mathematical properties relevant to quantum field theory.

Abstract

In these lectures I discuss Feynman graphs and the associated Feynman integrals. Of particular interest are the classes functions, which appear in the evaluation of Feynman integrals. The most prominent class of functions is given by multiple polylogarithms. The algebraic properties of multiple polylogarithms are reviewed in the second part of these lectures. The final part of these lectures is devoted to Feynman integrals, which cannot be expressed in terms of multiple polylogarithms. Methods from algebraic geometry provide tools to tackle these integrals.