On Feynman graphs, matroids, and GKZ-systems

TL;DR

This paper investigates the algebraic and combinatorial structures underlying Feynman diagrams, demonstrating that certain hypergeometric systems associated with them have normal semigroups and identifying related matroids.

Contribution

It establishes the normality of A-hypergeometric systems for Feynman diagrams in Lee--Pomeransky form and explores their associated matroids, advancing the understanding of their algebraic properties.

Findings

A-hypergeometric systems for Feynman diagrams have normal semigroups in key cases.

Identification of relevant matroids related to Feynman diagrams.

Connections between hypergeometric systems and matroid theory are elucidated.

Abstract

We show in several important cases that the $A$-hypergeometric system attached to a Feynman diagram in Lee--Pomeransky form, obtained by viewing the momenta and the nonzero masses as indeterminates, has a normal underlying semigroup. This continues a quest initiated by Klausen, and studied by Helmer and Tellander. In the process we identify several relevant matroids related to the situation and explore their relationships.