# Differential cohomology and locally covariant quantum field theory

June 05, 2014

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.

Keywords:

Abelian gauge theory, locally covariant quantum field theory, differential cohomology