Feynman graphs and Hyperplane arrangements defined 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.

5 pages, ver 2: added remarks on dual matroids, to appear in Journal of Geometry and Physics

References in corpus (1)

• Quantum field theory over F1