Differential cohomology and locally covariant quantum field theory

Christian Becker, Alexander Schenkel, Richard Joseph Szabo

Research output: Contribution to journalArticle

6 Citations (Scopus)

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 (Formula presented.)-functor, which yields a covariant functor from our categories of globally hyperbolic Lorentzian manifolds to the category of (Formula presented.)-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 Fréchet–Lie group structure on differential cohomology groups.

Original language English 1750003 Reviews in Mathematical Physics 29 1 7 Dec 2016 https://doi.org/10.1142/S0129055X17500039 Published - Feb 2017

Keywords

• Algebraic quantum field theory
• differential cohomology
• generalized Abelian gauge theory

ASJC Scopus subject areas

• Statistical and Nonlinear Physics
• Mathematical Physics