Differential cohomology and locally covariant quantum field theory

TL;DR

This paper develops a covariant quantum field theory framework based on differential cohomology on Lorentzian manifolds, revealing topological effects that violate locality but preserve causality and the time-slice property.

Contribution

It introduces a new covariant functor from differential cohomology to C*-algebras that satisfies causality and time-slice axioms, but not locality, due to topological substructures.

Findings

The functor satisfies causality and time-slice axioms.

It violates the locality axiom due to topological subfunctors.

A Fréchet-Lie group structure on differential cohomology is developed.

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.