Feynman graphs and Hyperplane arrangements defined over $\mathbb{F}_1$

TL;DR

This paper characterizes when hyperplane arrangements associated with graph cycle spaces have an $\\mathbb{F}_1$-structure, showing it occurs precisely for Boolean arrangements with specific cycle basis properties.

Contribution

It proves that arrangements have an $\mathbb{F}_1$-structure if and only if they are Boolean, linking combinatorial properties of cycle bases to algebraic structures.

Findings

Arrangement has an $\mathbb{F}_1$-structure iff it is Boolean.

Cycle space arrangement is Boolean iff it has a basis of cycles with non-overlapping edges.

Characterization connects graph cycle bases to algebraic structures over $\mathbb{F}_1$.

Abstract

Motivated by some computations of Feynman integrals and certain conjectures on mixed Tate motives, Bejleri and Marcolli posed questions about the $\mathbb{F}_1$-structure (in the sense of torification) on the complement of a hyperplane arrangement, especially for an arrangement defined in the space of cycles of a graph. In this paper, we prove that an arrangement has an $\mathbb{F}_1$-structure if and only if it is Boolean. We also prove that the arrangement in the cycle space of a graph is Boolean if and only if the cycle space has a basis consisting of cycles such that any two of them do not share edges.