Quantum field theory over F_1

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 F_1-geometry based on torified-schemes. We first discuss some simple necessary conditions in terms of the 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 F_1-structure. By a similar argument we show that the moduli spaces of curves M_(0,n) admit an F_1-structure, thus answering a question of Manin. We also discuss conditions on hyperplane arrangements, a possible notion of embedded F_1-structure and its relation to Chern classes, and questions on Chern classes of varieties with regular torifications.

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