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Quantum field theory over [formula omitted]

Authors
Journal
Journal of Geometry and Physics
0393-0440
Publisher
Elsevier
Volume
69
Identifiers
DOI: 10.1016/j.geomphys.2013.03.002
Keywords
  • Field With One Element
  • Perturbative Quantum Field Theory
  • Graph Hypersurfaces And Configuration Spaces
  • Moduli Spaces Of Curves
  • Torified-Schemes
  • Grothendieck Ring Of Varieties
Disciplines
  • Mathematics
  • Physics

Abstract

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 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 F1-structure. By a similar argument we show that the moduli spaces of curves M̄0,n admit an F1-structure, thus answering a question of Manin. 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.

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