Quantum field theory over F1

TL;DR

This paper explores the geometry over the field with one element (F1) in the context of quantum field theory, focusing on algebraic varieties related to Feynman integrals and moduli spaces, using torified schemes.

Contribution

It introduces conditions for F1-structures on varieties relevant to quantum field theory, including blowup formulas and structures on configuration and moduli spaces.

Findings

Wonderful compactifications admit F1-structures

Moduli spaces of genus zero curves admit F1-structures

Conditions on hyperplane arrangements and Chern classes are discussed

Abstract

In this paper we discuss some questions about geometry over the field with one element, motivated by the properties of algebraic varieties that arise in perturbative quantum field theory. We follow the approach to F1-geometry based on torified schemes. We first discuss some simple necessary conditions in terms of Euler characteristic and classes in the Grothendieck ring, then we give a blowup formula for torified varieties and we show that the wonderful compactifications of the graph configuration spaces, that arise in the computation of Feynman integrals in position space, admit an F1-structure. By a similar argument we show that the moduli spaces of curves of genus zero with n marked points admit an F1-structure. We also discuss conditions on hyperplane arrangements, a possible notion of embedded F1-structure and its relation to Chern classes, and questions on Chern classes of varieties with regular torifications.