Differential cohomology and locally covariant quantum field theory

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 Fréchet-Lie group structure on differential cohomology groups.

35 pages. v2: Final version published in Reviews in Mathematical Physics

References in corpus (6)

• The Uncertainty of Fluxes
• Heisenberg Groups and Noncommutative Fluxes
• Green-hyperbolic operators on globally hyperbolic spacetimes
• A C*-algebra for quantized principal U(1)-connections on globally hyperbolic Lorentzian manifolds
• Dynamical locality of the free Maxwell field
• Locally covariant quantum field theory with external sources

Cited by in corpus (15)

• QFT on curved spacetimes: axiomatic framework and examples
• Homotopical Quantum Field Theory
• Homotopy colimits and global observables in Abelian gauge theory
• Locally covariant quantum field theory and the problem of formulating the same physics in all spacetimes
• The Calabi complex and Killing sheaf cohomology
• Linear Yang-Mills theory as a homotopy AQFT
• Higher Structures in Algebraic Quantum Field Theory
• Quantum field theories on categories fibered in groupoids
• Snowmass White Paper: The Quest to Define QFT
• Hadamard states for quantum Abelian duality
• Group-valued momentum maps for actions of automorphism groups
• Locality in Abelian gauge theories over globally hyperbolic spacetimes
• Cheeger-Simons differential characters with compact support and Pontryagin duality
• Realizing the Teichmüller space as a symplectic quotient
• Aspects of Higher-Abelian Gauge Theories at zero and finite temperature: Topological Casimir effect, duality and Polyakov loops