Quantum fields, periods and algebraic geometry

TL;DR

This paper explores the intersection of graph theory, algebraic geometry, and quantum field theory, focusing on how graph polynomials relate to algebraic structures and the emergence of renormalization-independent periods.

Contribution

It introduces a framework connecting graph polynomials with algebraic geometry to analyze Feynman rules and renormalization periods in quantum field theory.

Findings

Identification of algebraic structures underlying Feynman graphs

Analysis of graph polynomials' role in algebraic geometry

Discussion of renormalization scheme independent periods

Abstract

We discuss how basic notions of graph theory and associated graph polynomials define questions for algebraic geometry, with an emphasis given to an analysis of the structure of Feynman rules as determined by those graph polynomials as well as algebraic structures of graphs. In particular, we discuss the appearance of renormalization scheme independent periods in quantum field theory.