Differential cohomology and locally covariant quantum field theory

Becker, Christian; Schenkel, Alexander; Szabo, Richard J.

Abstract

We study differential cohomology on categories of globally hyperbolic Lorentzian manifolds. The Lorentzian metric allows us to define a natural transformation whose kernel generalizes Maxwell's equations and fits into a restriction of the fundamental exact sequences of differential cohomology. We consider smooth Pontryagin duals of differential cohomology groups, which are subgroups of the character groups. We prove that these groups fit into smooth duals of the fundamental exact sequences of differential cohomology and equip them with a natural presymplectic structure derived from a generalized Maxwell Lagrangian. The resulting presymplectic Abelian groups are quantized using the CCR-functor, which yields a covariant functor from our categories of globally hyperbolic Lorentzian manifolds to the category of C*-algebras. We prove that this functor satisfies the causality and time-slice axioms of locally covariant quantum field theory, but that it violates the locality axiom. We show that this violation is precisely due to the fact that our functor has topological subfunctors describing the Pontryagin duals of certain singular cohomology groups. As a byproduct, we develop a Fr\'echet-Lie group structure on differential cohomology groups.

Citation

Becker, C., Schenkel, A., & Szabo, R. J. (2017). Differential cohomology and locally covariant quantum field theory. Reviews in Mathematical Physics, 29(1), Article 1750003. https://doi.org/10.1142/S0129055X17500039

• Journal Article Type: Article
• Acceptance Date: Nov 13, 2016
• Online Publication Date: Dec 7, 2016
• Publication Date: 2017-02
• Deposit Date: Mar 6, 2017
• Publicly Available Date: Mar 6, 2017
• Journal: Reviews in Mathematical Physics
• Print ISSN: 0129-055X
• Electronic ISSN: 1793-6659
• Publisher: World Scientific
• Peer Reviewed: Peer Reviewed
• Volume: 29
• Issue: 1
• Article Number: 1750003
• DOI: https://doi.org/10.1142/S0129055X17500039
• Keywords: algebraic quantum field theory, generalized Abelian gauge theory, differential cohomology
• Public URL: https://nottingham-repository.worktribe.com/output/843439
• Publisher URL: http://www.worldscientific.com/doi/abs/10.1142/S0129055X17500039
• Related Public URLs: https://arxiv.org/abs/1406.1514
• Additional Information: Electronic version of an article published as Reviews in Mathematical Physics, Volume 29, Issue 1, 2017, 1750003. doi:10.1142/S0129055X17500039 © copyright World Scientific Publishing Company. http://www.worldscientific.com/doi/abs/10.1142/S0129055X17500039