Differential cohomology and locally covariant quantum field theory

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.


Introduction and summary
In [CS85], Cheeger and Simons develop the theory of differential characters, which can be understood as a differential refinement of singular cohomology H k ( · ; Z). For a smooth manifold M , a differential character is a group homomorphism h : Z k−1 (M ; Z) → T from smooth singular k−1-cycles to the circle group T = U (1), which evaluated on smooth singular k−1-boundaries is given by integrating a differential form, the curvature curv(h) of h. This uniquely defines the curvature map curv : H k (M ; Z) → Ω k Z (M ), which is a natural group epimorphism from the group of differential characters to the group of k-forms with integral periods. To each differential character one can assign its characteristic class via a second natural group epimorphism char : H k (M ; Z) → H k (M ; Z), which is why one calls differential characters a differential refinement of H k (M ; Z). In addition to their characteristic class and curvature, differential characters carry further information that is described by two natural group monomorphisms ι : Ω k−1 (M )/Ω k−1 Z (M ) → H k (M ; Z) and κ : H k−1 (M ; T) → H k (M ; Z), which map, respectively, to the kernel of char and curv. The group of differential characters H k (M ; Z) together with these group homomorphisms fits into a natural commutative diagram of exact sequences, see e.g. (2.11) in the main text. It was recognized later in [SS08,BB13] that this diagram uniquely fixes (up to a unique natural isomorphism) the functors H k ( · ; Z). It is therefore natural to abstract these considerations and to define a differential cohomology theory as a contravariant functor from the category of smooth manifolds to the category of Abelian groups that fits (via four natural transformations) into the diagram (2.11).
Differential cohomology finds its physical applications in field theory and string theory as an efficient way to describe the gauge orbit spaces of generalized or higher Abelian gauge theories. The degree k = 2 differential cohomology group H 2 (M ; Z) describes isomorphism classes of pairs (P, ∇) consisting of a T-bundle P → M and a connection ∇ on P . Physically this is exactly the gauge orbit space of Maxwell's theory of electromagnetism. The characteristic class map char : 2π i d log h; this field theory is called the σ-model on M with target space T. In degree k ≥ 3, the differential cohomology groups H k (M ; Z) describe isomorphism classes of k−2-gerbes with connection, which are models of relevance in string theory; see e.g. [Sza12] for a general introduction.
The goal of this paper is to understand the classical and quantum field theory described by a differential cohomology theory. Earlier approaches to this subject [FMS07a,FMS07b] have focused on the Hamiltonian approach, which required the underlying Lorentzian manifold M to be ultrastatic, i.e. that the Lorentzian metric g on M = R × Σ M is of the form g = −dt ⊗ dt + h, where h is a complete Riemannian metric on Σ M that is independent of time t. Here we shall instead work in the framework of locally covariant quantum field theory [BFV03,FV12], which allows us to treat generic globally hyperbolic Lorentzian manifolds M without this restriction. In addition, our construction of the quantum field theory is functorial in the sense that we shall obtain a covariant functor A k ( · ) : Loc m → C * Alg from a suitable category of m-dimensional globally hyperbolic Lorentzian manifolds to the category of C * -algebras, which describes quantized observable algebras of a degree k differential cohomology theory. This means that in addition to obtaining for each globally hyperbolic Lorentzian manifold M a C * -algebra of observables A k (M ), we get C * -algebra morphisms A k (f ) : A k (M ) → A k (N ) whenever there is a causal isometric embedding f : M → N . This in particular provides a mapping of observables from certain subregions of M to M itself, which is known to encode essential physical characteristics of the quantum field theory since the work of Haag and Kastler [HK64].
Let us outline the content of this paper: In Section 2 we give a short introduction to differential cohomology, focusing both on the abstract approach and the explicit model of Cheeger-Simons differential characters. In Section 3 we restrict any (abstract) degree k differential cohomology theory to a suitable category of m-dimensional globally hyperbolic Lorentzian manifolds Loc m , introduce generalized Maxwell maps and study their solution subgroups (generalizing Maxwell's equations in degree k = 2). The solution subgroups are shown to fit into a fundamental commutative diagram of exact sequences. We also prove that local generalized Maxwell solutions (i.e. solutions given solely in a suitable region containing a Cauchy surface) uniquely extend to global ones. In Section 4 we study the character groups of the differential cohomology groups. Inspired by [HLZ03] we introduce the concept of smooth Pontryagin duals, which are certain subgroups of the character groups, and prove that they fit into a commutative diagram of fundamental exact sequences. We further show that the smooth Pontryagin duals separate points of the differential cohomology groups and that they are given by a covariant functor from Loc m to the category of Abelian groups. In Section 5 we equip the smooth Pontryagin duals with a natural presymplectic structure, which we derive from a generalized Maxwell Lagrangian by adapting Peierls' construction [Pei52]. This then leads to a covariant functor G k ( · ) from Loc m to the category of presymplectic Abelian groups, which describes the classical field theory associated to a differential cohomology theory. The generalized Maxwell equations are encoded by taking a quotient of this functor by the vanishing subgroups of the solution subgroups. Due to the fundamental commutative diagram of exact sequences for the smooth Pontryagin duals, we observe immediately that the functor G k ( · ) has two subfunctors, one of which is H k ( · ; Z) , the Pontryagin dual of Z-valued singular cohomology, and hence is purely topological. The second subfunctor describes "curvature observables" and we show that it has a further subfunctor H m−k ( · ; R) . This gives a more direct and natural perspective on the locally covariant topological quantum fields described in [BDS13] for connections on a fixed T-bundle. In Section 6 we carry out the canonical quantization of our field theory by using the CCR-functor for presymplectic Abelian groups developed in [M + 73] and also in [BDHS13,Appendix A]. This yields a covariant functor A k ( · ) : Loc m → C * Alg to the category of C *algebras. We prove that A k ( · ) satisfies the causality axiom and the time-slice axiom, which have been proposed in [BFV03] to single out physically reasonable models for quantum field theory from all covariant functors Loc m → C * Alg. The locality axiom, demanding that A k (f ) is injective for any Loc m -morphism f , is in general not satisfied (except in the special case (m, k) = (2, 1)). We prove that for a Loc m -morphism f : M → N the morphism A k (f ) is injective if and only if the morphism H m−k (M ; R) ⊕ H k (M ; Z) → H m−k (N ; R) ⊕ H k (N ; Z) in the topological subtheories is injective, which is in general not the case. This provides a precise connection between the violation of the locality axiom and the presence of topological subtheories, which generalizes the results obtained in [BDHS13] for gauge theories of connections on fixed T-bundles. In Appendix A we develop a Fréchet-Lie group structure on differential cohomology groups, which is required to make precise our construction of the presymplectic structure.

Differential cohomology
In this section we summarize some background material on (ordinary) differential cohomology that will be used in this paper. In order to fix notation we shall first give a condensed summary of singular homology and cohomology. We shall then briefly review the Cheeger-Simons differential characters defined in [CS85]. The group of differential characters is a particular model of (ordinary) differential cohomology. Even though our results in the ensuing sections are formulated in a model independent way, it is helpful to have the explicit model of differential characters in mind.

Singular homology and cohomology
Let M be a smooth manifold. We denote by C k (M ; Z) the free Abelian group of smooth singular k-chains in M . There exist boundary maps ∂ k : C k (M ; Z) → C k−1 (M ; Z), which are homomorphisms of Abelian groups satisfying ∂ k−1 • ∂ k = 0. The subgroup Z k (M ; Z) := Ker ∂ k is called the group of smooth singular k-cycles and it has the obvious subgroup B k (M ; Z) := Im ∂ k+1 of smooth singular k-boundaries. The k-th smooth singular homology group of M is defined as the quotient (2.1) Notice that H k ( · ; Z) : Man → Ab is a covariant functor from the category of smooth manifolds to the category of Abelian groups; for a Man-morphism f : M → N (i.e. a smooth map) the Abmorphism H k (f ; Z) : H k (M ; Z) → H k (N ; Z) is given by push-forward of smooth k-simplices.
In the following we shall often drop the adjective smooth singular and simply use the words k-chain, k-cycle and k-boundary. Furthermore, we shall drop the label k on the boundary maps ∂ k whenever there is no risk of confusion.
Given any Abelian group G, the Abelian group of G-valued k-cochains is defined by where Hom denotes the group homomorphisms. The boundary maps ∂ k dualize to the coboundary maps δ k : , which are homomorphisms of Abelian groups and satisfy δ k+1 •δ k = 0. Elements in Z k (M ; G) := Ker δ k are called G-valued k-cocycles and elements in B k (M ; G) := Im δ k−1 are called G-valued k-coboundaries. The (smooth singular) cohomology group with coefficients in G is defined by Notice that H k ( · ; G) : Man → Ab is a contravariant functor.
The cohomology group H k (M ; G) is in general not isomorphic to Hom(H k (M ; Z), G). The obvious group homomorphism H k (M ; G) → Hom(H k (M ; Z), G) is in general only surjective but not injective. Its kernel is described by the universal coefficient theorem for cohomology (see e.g. [Hat02, Theorem 3.2]), which states that there is an exact sequence In this paper the group G will be either Z, R or T = U (1) (the circle group). As R and T are divisible groups, we have Ext( · , R) = Ext( · , T) = 0. Thus

Differential characters
Let M be a smooth manifold and denote by Ω k (M ) the R-vector space of smooth k-forms on M .
Definition 2.1. The Abelian group of degree k differential characters 1 on M , with 1 ≤ k ∈ N, is defined by (2.5) By the notation h • ∂ ∈ Ω k (M ) we mean that there exists ω h ∈ Ω k (M ) such that The Abelian group structure on H k (M ; Z) is defined pointwise. As it will simplify the notations throughout this paper, we shall use an additive notation for the group structure on H k (M ; Z), even though this seems counterintuitive from the perspective of differential characters. Explicitly, we define the group operation + on H k (M ; Z) by (h + l)(z) There are various interesting group homomorphisms with the Abelian group H k (M ; Z) as target or source. The first one is obtained by observing that the form ω h ∈ Ω k (M ) in (2.6) is uniquely determined for any h ∈ H k (M ; Z). Furthermore, ω h is closed, i.e. dω h = 0 with d being the exterior differential, and it has integral periods, i.e. z ω h ∈ Z for all z ∈ Z k (M ; Z). We denote the Abelian group of closed k-forms with integral periods by Ω k is the subspace of closed k-forms. Hence we have found a group homomorphism which we call the curvature.
We can also associate to each differential character its characteristic class, which is an element in H k (M ; Z). There exists a group homomorphism called the characteristic class, which is constructed as follows: Since Z k−1 (M ; Z) is a free Z-module, any h ∈ H k (M ; Z) has a real lifth ∈ Hom(Z k−1 (M ; Z), R) such that h(z) = exp(2π ih(z)) for all z ∈ Z k−1 (M ; Z). We define a real valued k-cochain by µh : . It can be easily checked that µh is a k-cocycle, i.e. δµh = 0, and that it takes values in Z. We define the class char(h) := [µh] ∈ H k (M ; Z) and note that it is independent of the choice of lifth of h.
It can be shown that the curvature and characteristic class maps are surjective, however, in general they are not injective [CS85]. This means that differential characters have further properties besides their curvature and characteristic class. In order to characterize these properties we shall define two further homomorphisms of Abelian groups with H k (M ; Z) as target: Firstly, the topological trivialization is the group homomorphism ι : . This expression is well-defined since by definition z η ∈ Z for all η ∈ Ω k−1 Z (M ) and z ∈ Z k−1 (M ; Z). Secondly, the inclusion of flat classes is the group homomorphism As shown in [CS85,BB13], the various group homomorphisms defined above fit into a commutative diagram of short exact sequences.
Theorem 2.2. The following diagram of homomorphisms of Abelian groups commutes and its rows and columns are exact sequences: The Abelian group of differential characters H k (M ; Z), as well as all other Abelian groups appearing in the diagram (2.11), are given by contravariant functors from the category of smooth manifolds Man to the category of Abelian groups Ab. The morphisms appearing in the diagram (2.11) are natural transformations.
Example 2.3. The Abelian groups of differential characters H k (M ; Z) can be interpreted as gauge orbit spaces of (higher) Abelian gauge theories, see e.g. [BB13, Examples 5.6-5.8] for mathematical details and [Sza12] for a discussion of physical applications.
1. In degree k = 1, the differential characters 2. In degree k = 2, the differential characters H 2 (M ; Z) describe isomorphism classes of T-bundles with connections (P, ∇) on M . The holonomy map associates to any onecycle z ∈ Z 1 (M ; Z) a group element h(z) ∈ T. This defines a differential character h ∈ H 2 (M ; Z), whose curvature is curv(h) = − 1 2π i F ∇ and whose characteristic class is the first Chern class of P . The topological trivialization ι : 3. In degree k ≥ 3, the differential characters H k (M ; Z) describe isomorphism classes of k−2-gerbes with connections, see e.g. [Hit01] for the case of usual gerbes, i.e. k = 3. These models are examples of higher Abelian gauge theories where the curvature is given by a k-form, and they physically arise in string theory, see e.g. [Sza12].

Differential cohomology theories
The functor describing Cheeger-Simons differential characters is a specific model of what is called a differential cohomology theory. There are also other explicit models for differential cohomology, as for example those obtained in smooth Deligne cohomology (see e.g. [Sza12]), the de Rham-Federer approach [HLZ03] making use of de Rham currents (i.e. distributional differential forms), and the seminal Hopkins-Singer model [HS05] which is based on differential cocycles and the homotopy theory of differential function spaces. These models also fit into the commutative diagram of exact sequences in (2.11). The extent to which (2.11) determines the functors H k ( · ; Z) has been addressed in [SS08,BB13] and it turns out that they are uniquely determined (up to a unique natural isomorphism). This motivates the following Theorem 2.5 ([BB13, Theorems 5.11 and 5.14]). For any differential cohomology theory ( H * ( · ; Z), curv, char, ι, κ) there exists a unique natural isomorphism Ξ : H * ( · ; Z) ⇒ H * ( · ; Z) to differential characters such that Remark 2.6. In order to simplify the notation we shall denote in the following any differential cohomology theory by ( H * ( · ; Z), curv, char, ι, κ).

Generalized Maxwell maps
Our main interest lies in understanding the classical and quantum field theory described by a differential cohomology theory H * ( · ; Z) : Man → Ab Z . For a clearer presentation we shall fix 1 ≤ k ∈ Z and study the differential cohomology groups of degree k, i.e. the contravariant functor H k ( · ; Z) : Man → Ab. Furthermore, in order to formulate relativistic field equations which generalize Maxwell's equations in degree k = 2, we shall restrict the category of smooth manifolds to a suitable category of globally hyperbolic spacetimes. A natural choice, see e.g. [BFV03,FV12,BG11,BGP07], is the following Definition 3.1. The category Loc m consists of the following objects and morphisms: • The objects M in Loc m are oriented and time-oriented globally hyperbolic Lorentzian manifolds, which are of dimension m ≥ 2 and of finite-type. 2 (For ease of notation we shall always suppress the orientation, time-orientation and Lorentzian metric.) •  When working on the category Loc m we have available a further natural transformation given by the codifferential δ : Ω * ( · ) ⇒ Ω * −1 ( · ). 3 Our conventions for the codifferential δ are as follows: Denoting by * the Hodge operator, we define δ on p-forms by For any two forms ω, ω ∈ Ω p (M ) with compactly overlapping support we have a natural indefinite inner product defined by Then the codifferential δ is the formal adjoint of the differential d with respect to this inner product, i.e. δω, ω = ω, dω for all ω ∈ Ω p (M ) and ω ∈ Ω p−1 (M ) with compactly overlapping support.  The next goal is to restrict the diagram (2.12) to the solution subgroup Sol k (M ) ⊆ H k (M ; Z). Let us denote by Ω k Z, δ (M ) the Abelian group of closed and coclosed k-forms with integral periods. From the definition of the solution subgroups (3.4) it is clear that the middle horizontal sequence in (2.12) restricts to the exact sequence In order to restrict the complete diagram (2.12) to the solution subgroups we need the following Lemma 3.6. The inverse image of Sol k (M ) under the topological trivialization ι is given by Denoting by (dΩ k−1 ) δ (M ) the space of exact k-forms which are also coclosed, we obtain Theorem 3.7. The following diagram commutes and has exact rows and columns: Proof. The only nontrivial step is to show that char : be any cohomology class. By the middle vertical exact sequence in (2.12) there is again u as ι maps to the kernel of char. We now show that [η] can be chosen such that MW(h ) = 0, which completes the proof. By posing MW(h ) = 0 as a condition we obtain the partial differential equation Remark 3.8. In the context of compact Riemannian manifolds, a result similar to Theorem 3.7 is proven in [GM09]. They consider harmonic differential characters on a compact Riemannian manifold, i.e. differential characters with harmonic curvature forms, and prove that these fit into exact sequences similar to the ones in (3.7). However, the proof in [GM09] relies on the theory of elliptic partial differential equations and therefore differs from our proof of Theorem 3.7, which uses the theory of hyperbolic partial differential equations. In particular, the results of [GM09] do not imply our results.
We say that a We first prove injectivity. Let h ∈ Sol k (N ) be any element in the kernel of Sol k (ι N ;O ). Applying char implies that char(h) lies in the kernel of which is an isomorphism since O and N are both homotopic to their common Cauchy surface. As a consequence char(h) = 0 and by Theorem 3.7 there exists [η] ∈ Sol k (N ) such that h = ι [η] . Since ι is natural and injective, this implies that [η] lies in the kernel of Sol k (ι N ;O ). We can always choose a coclosed representative η ∈ Ω k−1 δ (N ) of the class [η] (by going to Lorenz gauge, cf. [SDH14, Section 2.3]) and the condition that [η] lies in the kernel of [BGP07,Bär13]. Since O ⊆ N contains a Cauchy surface of N , any form of timelike compact support on O can be extended by zero to a form of timelike compact support on N (denoted with a slight abuse of notation by the same symbol). Hence there exists ρ ∈ Ω k−1 (N ) satisfying ρ| O = 0 such that η = G(δα + dβ) + ρ on all of N . As η satisfies δdη = 0 and the Lorenz gauge condition δη = 0, it also satisfies η = 0. Since also G(δα + dβ) = 0, we obtain ρ = 0, which together with the support condition ρ| O = 0 implies ρ = 0. So η = G(δα+dβ) on all of N and it remains to prove that η ∈ Ω k−1 Z (N ). As η is obviously closed, the integral z η depends only on the homology class We now prove surjectivity. Let l ∈ Sol k (O) be arbitrary and consider its characteristic class char(l) ∈ H k (O; Z). As we have explained above, is an isomorphism, hence by using also Theorem 3.7 we can find h ∈ Sol k (N ) such that . We can extend α by zero (denoted with a slight abuse of notation by the same symbol) and , which gives the assertion.

Smooth Pontryagin duality
Let H * ( · ; Z), curv, char, ι, κ be a differential cohomology theory and let us consider its restriction H k ( · ; Z) : Loc m → Ab to degree k ≥ 1 and to the category Loc m with m ≥ k. For an Abelian group G, the character group is defined by G := Hom(G, T), where Hom denotes the homomorphisms of Abelian groups. Since the circle group T is divisible, the Hom-functor Hom( · , T) preserves exact sequences. Hence we can dualize the degree k component of the diagram (2.12) and obtain the following commutative diagram with exact rows and columns: The diagram (4.1) contains the character groups of H k (M ; Z) and of various groups of differential forms, whose generic elements are too singular for our purposes. We shall use a strategy similar to [HLZ03] (called smooth Pontryagin duality) in order to identify suitable subgroups of such character groups, which describe regular group characters. In order to explain the construction of the smooth Pontryagin duals of the Abelian group H k (M ; Z) and the various groups of differential forms, let us first notice that there exists an injective homomorphism of Abelian groups W : Hence we have to understand the vanishing subgroups of differential forms with integral periods, Proof. We first show the inclusion "⊇": Let ϕ ∈ Ω p 0 (M ) be coclosed, i.e. δϕ = 0, and such that [ϕ], · restricts to a homomorphism of Abelian groups H p where in the second equality we have used the fact that the pairing depends only on the equivalence classes and in the last equality we have used [ω] ∈ H p free (M ; Z) via the de Rham isomorphism.
Let us now show the inclusion "⊆": Motivated by the definition (4.3) we define the smooth Pontryagin dual of the quotient We need to prove that η is closed and has integral periods, which implies that . As H k−1 free (M ; Z) is finitely generated (by our assumption that M is of finite-type) and free, its double dual Z-module is isomorphic to itself, hence the class [η] defines an element in H k−1 free (M ; Z) and as a consequence η has integral periods.
Using further the natural isomorphism (see e.g. [HLZ03, Lemma 5.1]) we observe that the restriction of the lowest row of the diagram (4.1) to smooth Pontryagin duals reads as with the dual group homomorphisms (4.9b) Here we have implicitly used the injective homomorphism of Abelian groups W : We now define the smooth Pontryagin dual of the differential cohomology group H k (M ; Z) by the inverse image (4.11) , as in this way we divide out from the smooth group characters on Ω k (M ) exactly those which are trivial on Ω k Z (M ). The diagram (4.1) restricts as follows to the smooth Pontryagin duals. Theorem 4.3. The following diagram commutes and has exact rows and columns: Proof. By the constructions above and (4.1), we have the following commutative subdiagram with exact rows and columns: and it remains to prove that it extends to the diagram of exact sequences in (4.12).
Let us first focus on the left column in (4.12). By (4.1), there exists an injective group homomorphism H k free (M ; Z) → Ω k Z (M ) and we have to show that its image lies in the smooth (4.14) In the last equality we have used the fact that, by Poincaré duality and de Rham's theorem, . Exactness of the corresponding sequence (the left column in (4.12)) is an easy check.
It remains to understand the middle horizontal sequence in (4.12). From the commutative square in the lower left corner of (4.1) and the definition (4.11), we find that curv : Ω k Z (M ) → H k (M ; Z) restricts to the smooth Pontryagin duals: by commutativity of this square, ι • curv maps the smooth Pontryagin dual Ω k by the definition (4.11). We therefore get the middle horizontal sequence in (4.12) and it remains to prove that it is exact everywhere. As the restriction of an injective group homomorphism, curv :  for any Loc m -morphism f : M → N , and for all ϕ ∈ V p (M ) and ω ∈ Ω p Z (N ), where f * denotes the pull-back of differential forms. In the last equality we have used the fact that closed p-forms with integral periods on N are pulled-back under f to such forms on M . Thus in the diagram (4.12) we can regard Ω k 0 ( · )/V k ( · ), δΩ k 0 ( · ) and V k−1 ( · ) as covariant functors from Loc m to Ab. Furthermore, as a consequence of being the character groups (or dual Z-modules) of Abelian groups given by contravariant functors from Loc m to Ab, we can also regard H k free ( · ; Z) , H k ( · ; Z) , H k tor ( · ; Z) , H k−1 ( · ; T) and H k−1 free ( · ; Z) as covariant functors from Loc m to Ab. (Indeed, they are just given by composing the corresponding contravariant functors of degree k in (2.12) with the contravariant Hom-functor Hom( · , T) in case of the character groups or with Hom( · , Z) in case of the dual Z-modules.) By the same argument, the full character groups H k ( · ; Z) of H k ( · ; Z) are given by a covariant functor H k ( · ; Z) : Loc m → Ab. Finally, (4.12) is a diagram of natural transformations since it is the restriction to smooth Pontryagin duals of the diagram (4.1) of natural transformations, which is given by acting with the Hom-functor Hom( · , T) on the degree k component of the natural diagram (2.12).
Remark 4.6. For any Loc m -morphism f : M → N we shall denote the restriction of

Presymplectic Abelian group functors
As a preparatory step towards the quantization of the smooth Pontryagin dual H k ( · ; Z) ∞ : Loc m → Ab of a degree k differential cohomology theory we have to equip the Abelian groups H k (M ; Z) ∞ with a natural presymplectic structure τ : H k (M ; Z) ∞ × H k (M ; Z) ∞ → R. A useful selection criterion for these structures is given by Peierls' construction [Pei52] that allows us to derive a Poisson bracket which can be used as a presymplectic structure on H k (M ; Z) ∞ . We shall now explain this construction in some detail, referring to [BDS13, Remark 3.5] where a similar construction is done for connections on a fixed T-bundle.
Let M be any object in Loc m . Recall that any element w ∈ H k (M ; Z) ∞ is a group character, i.e. a homomorphism of Abelian groups w : H k (M ; Z) → T to the circle group T. Using the inclusion T → C of the circle group into the complex numbers of modulus one, we may regard w as a complex-valued functional, i.e. w : H k (M ; Z) → C. We use the following notion of functional derivative, which we derive in Appendix A from a Fréchet-Lie group structure on H k (M ; Z).
Proof. We compute (5.1) explicitly to get In the first equality we have used the fact that w is a homomorphism of Abelian groups and in the second equality the group homomorphism (4.2).
To work out Peierls' construction we need a Lagrangian, which we take to be the generalized Maxwell Lagrangian where λ > 0 is a "coupling" constant and the factor 1 2 is purely conventional. The corresponding Euler-Lagrange equation coincides (up to the factor λ) with the Maxwell map defined in Section 3, so they have the same solution subgroups. Given any solution h ∈ Sol k (M ) of the Euler-Lagrange equation λ δ curv(h) = 0, Peierls' proposal is to study the retarded/advanced effect of a functional w on this solution. Adapted to our setting, we shall introduce a formal parameter ε and search for η ± w ∈ Ω k−1 (M ) such that h ± w := h + ι [ε η ± w ] solves the partial differential equation up to first order in ε and such that η ± w satisfies a suitable asymptotic condition to be stated below. Expanding (5.5) to first order in ε (and using δ curv(h) = 0) yields the inhomogeneous equation The requisite asymptotic condition on η ± w is as follows: There exist small gauge transformations dχ ± w ∈ dΩ k−2 (M ) and Cauchy surfaces Σ ± w in M such that where J ± M (A) denotes the causal future/past of a subset A ⊆ M . In simple terms, this requires η + w to be pure gauge in the far past and η − w to be pure gauge in the far future. Under these assumptions, the unique (up to small gauge invariance) solution to (5.6) is given by η The difference between the retarded and advanced effects defines a Poisson bracket on the associative, commutative and unital * -algebra generated by H k (M ; Z) ∞ . For two generators v, w ∈ H k (M ; Z) ∞ the Poisson bracket reads as with the antisymmetric bihomomorphism of Abelian groups In this expression G := G + − G − : Ω k−1 0 (M ) → Ω k−1 (M ) is the retarded-minus-advanced Green's operator. Antisymmetry of τ follows from the fact that G is formally skew-adjoint as a consequence of being formally self-adjoint with respect to the inner product on forms · , · . By naturality of the Green's operators G ± and the inner product · , · , the presymplectic structure τ is also natural. This allows us to promote the covariant functor H k ( · ; Z) ∞ : Loc m → Ab to a functor with values in the category of presymplectic Abelian groups PAb defined as follows: The objects in PAb are pairs (G, σ), where G is an Abelian group and σ : G × G → R is an antisymmetric bihomomorphism of Abelian groups (called a presymplectic structure), i.e. for any g ∈ G, the maps σ( · , g), σ(g, · ) : G → R are both homomorphisms of Abelian groups. The morphisms in PAb are group homomorphisms φ : G → G that preserve the presymplectic structures, i.e. σ • (φ × φ) = σ.
The terminology off-shell comes from the physics literature and it means that the Abelian groups underlying G k o ( · ) are (subgroups of) the character groups of H k ( · ; Z). In contrast, the Abelian groups underlying the on-shell presymplectic Abelian group functor should be (subgroups of) the character groups of the subfunctor Sol k ( · ) of H k ( · ; Z), see Section 3. We shall discuss the on-shell presymplectic Abelian group functor later in this section after making some remarks on G k o ( · ). Our first remark is concerned with the presymplectic structure (5.10). Notice that τ is the pull-back under ι of the presymplectic structure on the Abelian group V k−1 (M ) given by where the horizontal and vertical sequences are exact.
Remark 5.4. The diagram (5.13) has the following physical interpretation. If we think of the covariant functor G k o ( · ) as a field theory describing classical observables on the differential cohomology groups H k ( · ; Z), the diagram shows that this field theory has two (faithful) subtheories: The first subtheory H k ( · ; Z) is purely topological and it describes observables on the cohomology groups H k ( · ; Z). The second subtheory F k o ( · ) describes only the "field strength observables", i.e. classical observables measuring the curvature of elements in H k ( · ; Z). In addition to G k o ( · ) having two subtheories, it also projects onto the field theory G k o ( · ) describing classical observables of topologically trivial fields.
Remark 5.5. In the PAb-diagram (5.13) the character group H k−1 (M ; T) (cf. the Ab-diagram (4.12)) does not appear. The reason is that there is no presymplectic structure on H k−1 ( · ; T) such that the components of both curv and κ are PAb-morphisms: if such a presymplectic structure σ would exist, then the presymplectic structure on F k o ( · ) would have to be trivial as it would be given by the pull-back of σ along κ • curv = 0. This is not the case. We expect that the role of the flat classes H k−1 ( · ; T) is that of a local symmetry group of the field theory G k o ( · ). This claim is strengthened by noting that adding flat classes does not change the generalized Maxwell Lagrangian (5.4). In future work we plan to study this local symmetry group in detail and also try to understand its role in Abelian S-duality.
Definition 5.7. The on-shell presymplectic Abelian group functor G k ( · ) : Loc m → PAb for a degree k differential cohomology theory is defined as the quotient G k ( · ) := G k o ( · )/ I k ( · ). Explicitly, it associates to an object M in Loc m the presymplectic Abelian group We shall now derive the analog of the diagram (5.13) for on-shell functors. By construction, it is clear that we have two natural transformations char : H k ( · ; Z) ⇒ G k ( · ) and curv : , which however do not have to be injective. To make them injective we have to take a quotient by the kernel subfunctors, which we now characterize.
Lemma 5.8. For any object M in Loc m , the kernel of char : . Taking now the quotient of the covariant functor F k o ( · ) : Loc m → PAb by its subfunctor K k ( · ), we get another covariant functor F k ( · ) := F k o ( · )/K k ( · ) : Loc m → PAb. By Lemma 5.8 there are now two injective natural transformations char : H k ( · ; Z) ⇒ G k ( · ) and curv : F k ( · ) ⇒ G k ( · ) to the on-shell presymplectic Abelian group functor G k ( · ). To obtain the on-shell analog of the diagram (5.13) we just have to notice that, by a proof similar to that of Proposition 5.6, the vanishing subgroups of the topologically trivial field theory are given by We define the corresponding quotient G k ( · ) := G k o ( · )/I k ( · ) : Loc m → PAb and notice that the natural transformation ι : G k ( · ) ⇒ G k ( · ) is well-defined and surjective. Hence we obtain a natural diagram in the category PAb given by where the horizontal and vertical sequences are exact. The physical interpretation given in Remark 5.4 applies to this diagram as well.
We conclude this section by pointing out that the subtheory F k ( · ) of G k ( · ) has a further purely topological subtheory. Let M be any object in Loc m . Recall that the Abelian group underlying Remark 5.9. Following the terminology used in ordinary Maxwell theory (given in degree k = 2) we may call the subtheory H m−k ( · ; R) electric and the subtheory H k ( · ; Z) magnetic. The structures we have found for the on-shell field theory can be summarized by the following diagram with all horizontal and vertical sequences exact:

Quantum field theory
In the previous section we have derived various functors from the category Loc m to the category PAb of presymplectic Abelian groups. In particular, the functor S k ( · ) describes the association of the smooth Pontryagin duals (equipped with a natural presymplectic structure) of the solution subgroups of a degree k differential cohomology theory. To quantize this field theory, we shall make use of the CCR-functor for presymplectic Abelian groups, see [M + 73] and [BDHS13, Appendix A] for details. In short, canonical quantization is a covariant functor CCR( · ) : PAb → C * Alg to the category of unital C * -algebras with morphisms given by unital C * -algebra homomorphisms (not necessarily injective). To any presymplectic Abelian group (G, σ) this functor associates the unital C * -algebra CCR(G, σ), which is generated by the symbols W (g), g ∈ G, satisfying the Weyl relations W (g) W (g) = e − i σ(g,g)/2 W (g +g) and the * -involution property W (g) * = W (−g). This unital * -algebra is then equipped and completed with respect to a suitable C * -norm. To any PAb-morphism φ : (G, σ) → (G , σ ) the functor associates the C * Alg-morphism CCR(φ) : CCR(G, σ) → CCR(G , σ ), which is obtained as the unique continuous extension of the unital * -algebra homomorphism defined on generators by W (g) → W (φ(g)).
Definition 6.1. The quantum field theory functor A k ( · ) : Loc m → C * Alg for a degree k differential cohomology theory is defined as the composition of the on-shell presymplectic Abelian group functor G k ( · ) : Loc m → PAb with the CCR-functor CCR( · ) : PAb → C * Alg, i.e.
A k ( · ) := CCR( · ) • G k ( · ) . (6.1) Remark 6.2. The subtheory structure of the classical on-shell field theory explained in Remark 5.9 is also present, with a slight caveat, in the quantum field theory. Acting with the functor CCR( · ) on the diagram (5.30) we obtain a similar diagram in the category C * Alg (with CCR(0) = C, the trivial unital C * -algebra). However, the sequences in this diagram will in general not be exact, as the CCR-functor is not an exact functor. This will not be of major concern to us, since by [BDHS13, Corollary A.7] the functor CCR( · ) does map injective PAb-morphisms to injective C * Alg-morphisms. Thus our statements in Remark 5.9 about the (faithful) subtheories remain valid after quantization. Explicitly, the quantum field theory A k ( · ) has three faithful subtheories A k el ( · ) := CCR( · ) • H m−k ( · ; R) , A k mag ( · ) := CCR( · ) • H k ( · ; Z) and A k F ( · ) := CCR( · ) • F k ( · ), with the first two being purely topological and the third being a theory of quantized curvature observables.
We shall now address the problem of whether or not our functor A k ( · ) satisfies the axioms of locally covariant quantum field theory, which have been proposed in [BFV03] to single out physically reasonable models for quantum field theory from all possible covariant functors from Loc m to C * Alg. The first axiom formalizes the concept of Einstein causality.
Proof. For any two generators W (w) ∈ A k (M 1 ) and W (v) ∈ A k (M 2 ) we have where we have used the Weyl relations and the fact that, by hypothesis, the push-forwards f 1 * (ι (w)) and f 2 * (ι (v)) are differential forms of causally disjoint support, for which the presymplectic structure (5.11) vanishes. The result now follows by approximating generic elements in A k (M 1 ) and A k (M 2 ) by linear combinations of generators and using continuity of A k (f 1 ) and A k (f 2 ).
The second axiom formalizes the concept of a dynamical law. Recall that a Loc m -morphism f : M → N is called a Cauchy morphism if the image f [M ] contains a Cauchy surface of N .
Theorem 6.4. The functor A k ( · ) : Loc m → C * Alg satisfies the time-slice axiom: Proof. Recall that the Abelian groups underlying G k ( · ) are subgroups of the character groups of Sol k ( · ) and that by definition Using Theorem 3.9 we have that Sol k (f ) is an Ab-isomorphism for any Cauchy morphism f : M → N , hence G k (f ) is a PAb-isomorphism and as a consequence of functoriality A k (f ) = CCR G k (f ) is a C * Alg-isomorphism.
In addition to the causality and time-slice axioms, [BFV03] proposed the locality axiom which demands that the functor A k ( · ) : Loc m → C * Alg should map any Loc m -morphism f : M → N to an injective C * Alg-morphism A k (f ) : A k (M ) → A k (N ). The physical idea behind this axiom is that any observable quantity on a sub-spacetime M should also be an observable quantity on the full spacetime N into which it embeds via f : M → N . It is known that this axiom is not satisfied in various formulations of Maxwell's theory, see e.g. [DL12,BDS13,BDHS13,SDH14,FL14]. The violation of the locality axiom is shown in most of these works by giving an example of a Loc m -morphism f : M → N such that the induced C * -algebra morphism is not injective. A detailed characterization and understanding of which Loc m -morphisms violate the locality axiom is given in [BDHS13] for a theory of connections on fixed T-bundles. It is shown there that a morphism violates the locality axiom if and only if the induced morphism between the compactly supported de Rham cohomology groups of degree 2 is not injective. Thus the locality axiom is violated due to topological obstructions. Our present theory under consideration has a much richer topological structure than a theory of connections on a fixed T-bundle, see Remark 6.2. It is therefore important to extend the analysis of [BDHS13] to our functor A k ( · ) : Loc m → C * Alg in order to characterize exactly those Loc m -morphisms which violate the locality axiom.
We collect some results which will simplify our analysis: Let φ : (G, σ) → (G , σ ) be any PAb-morphism. Then CCR(φ) is injective if and only if φ is injective: the direction "⇐" is shown in [BDHS13,Corollary A.7] and the direction "⇒" is an obvious proof by contraposition (which is spelled out in [BDHS13, Theorem 5.2]). Hence our problem of characterizing all Loc mmorphisms f : M → N for which A k (f ) is injective is equivalent to the classical problem of characterizing all Loc m -morphisms f : M → N for which G k (f ) is injective. Furthermore, the kernel of any PAb-morphism φ : (G, σ) → (G , σ ) is a subgroup of the radical in (G, σ): if g ∈ G with φ(g) = 0 then 0 = σ φ(g), φ(g) = σ(g,g) for allg ∈ G, which shows that g is an element the radical of (G, σ). For any object M in Loc m the radical of G k (M ) is easily computed: Lemma 6.5. The radical of G k (M ) is the subgroup Proof. We show the inclusion "⊇" by evaluating the presymplectic structure (5.10) for any element v of the group on the right-hand side of (6.3) and any w ∈ G k (M ). Using ι (v) = [δdρ] for some ρ ∈ Ω k−1 tc (M ), we obtain We now show the inclusion "⊆". Let v be any element in the radical of G k (M ), i.e. 0 = τ (w, v) = λ −1 ι (w), G(ι (v)) for all w ∈ G k (M ). As ι is surjective, this implies that ϕ, G(ι (v)) = 0 for all ϕ ∈ V k−1 (M ), from which we can deduce by similar arguments as in the proof of Proposition 5.6 that ι (v) = [δdρ] for some ρ ∈ Ω k−1 tc (M ) with dρ ∈ Ω k 0 (M ).
We show that the radical Rad G k (M ) , and hence also the kernel of any PAb-morphism with source given by G k (M ), is contained in the images of curv • q : H m−k (M ; R) → G k (M ) and char : H k (M ; Z) → G k (M ). To make this precise, similarly to [FS14, Section 5] we may equip the category PAb with the following monoidal structure ⊕: For two objects (G, σ) and (G , σ ) in PAb we set (G, σ) ⊕ (G , σ ) := (G ⊕ G , σ ⊕ σ ), where G ⊕ G denotes the direct sum of Abelian groups and σ ⊕ σ is the presymplectic structure on G ⊕ G defined by σ ⊕ σ g ⊕ g ,g ⊕g := σ(g,g) + σ (g ,g ). For two PAb-morphisms φ i : (G i , σ i ) → (G i , σ i ), i = 1, 2, the functor gives the direct sum φ 1 ⊕ φ 2 : (G 1 ⊕ G 2 , σ 1 ⊕ σ 2 ) → (G 1 ⊕ G 2 , σ 1 ⊕ σ 2 ). The identity object is the trivial presymplectic Abelian group. We define the covariant functor describing the direct sum of both topological subtheories of G k ( · ) by There is an obvious natural transformation top : Charge k ( · ) ⇒ G k ( · ) given for any object M in Loc m by We can now give a characterization of the Loc m -morphisms which violate the locality axiom.
Theorem 6.8. Let f : M → N be any Loc m -morphism. Then the C * Alg-morphism A k (f ) : Proof. We can simplify this problem by recalling from above that A k (f ) is injective if and only if G k (f ) is injective. Furthermore, it is easier to prove the contraposition " G k (f ) not injective ⇔ Charge k (f ) not injective", which is equivalent to our theorem. Our arguments will be based on the fact that top : Charge k ( · ) ⇒ G k ( · ) is an injective natural transformation, so it is helpful to draw the corresponding commutative diagram in the category PAb with exact vertical sequences: Let us prove the direction "⇐": Assuming that Charge k (f ) is not injective, the diagram (6.7) implies that top • Charge k (f ) = G k (f ) • top is not injective, and hence G k (f ) is not injective since top is injective. To prove the direction "⇒" let us assume that G k (f ) is not injective. By Lemma 6.6 the kernel of G k (f ) is a subgroup of the image of Charge k (M ) under top : Charge k (M ) → G k (M ), hence G k (f ) • top is not injective. The commutative diagram (6.7) then implies that top • Charge k (f ) is not injective, hence Charge k (f ) is not injective since top is injective. • k = 1: Since by assumption m ≥ 2, the second isomorphism in (6.8) implies Charge 1 (N ) = 0. Since 1 ≤ p ≤ m − 1, we have H m−1 (S p−1 ; R) = 0 and hence the first isomorphism in (6.8) implies Charge 1 (M ) H 1 (S p−1 ; Z) . For m ≥ 3 we choose p = 2 and find that Charge 1 (M ) Z T, hence Charge 1 (ι N ;M ) is not injective (being a group homomorphism T → 0). The case m = 2 is special and is discussed in detail below.
• 2 ≤ k ≤ m − 1: The second isomorphism in (6.8) implies Charge k (N ) = 0. Choosing p = m − k + 1 (which is admissible since 2 ≤ p ≤ m − 1), the first isomorphism in (6.8) gives Charge k (M ) R ⊕ δ k,m−k T, where δ k,m−k denotes the Kronecker delta. Hence Charge k (ι N ;M ) is not injective (being a group homomorphism R ⊕ δ k,m−k T → 0). Alternatively, if 2 ≤ k ≤ m − 2 we may also choose p = k + 1 and find via the first isomorphism in (6.8) that Charge k (M ) δ k,m−k R ⊕ T, which also implies that Charge k (ι N ;M ) is not injective.
• k = m: The second isomorphism in (6.8) implies Charge m (N ) R. Choosing p = 1 (which is admissible since m ≥ 2) we obtain Charge m (M ) R 2 , hence Charge m (ι N ;M ) is not injective (being a group homomorphism R 2 → R).
Proof. This follows from the explicit examples of Loc m -morphisms given in Example 6.9 and Theorem 6.8.
The case m = 2 and k = 1 is special. As any object M in Loc 2 is a two-dimensional globally hyperbolic spacetime, there exists a one-dimensional Cauchy surface Σ M such that M R × Σ M . By the classification of one-manifolds (without boundary), Σ M is diffeomorphic to the disjoint union of copies of R and T, i.e. Σ M T (the natural numbers n M and c M are finite, since M is assumed to be of finite-type). By homotopy invariance, we have As any Loc m -morphism f : M → N is in particular an embedding, the number of compact components in the Cauchy surfaces Σ M and Σ N cannot decrease, i.e. c M ≤ c N . As a consequence, Charge 1 (f ) is injective and by Theorem 6.8 so is A 1 (f ).
Proposition 6.11. The quantum field theory functor A 1 ( · ) : Loc 2 → C * Alg satisfies the locality axiom. Thus it is a locally covariant quantum field theory in the sense of [BFV03]. gebraic Quantum Field Theory: Its Status and its Future", where we received useful feedback and suggestions on this work from the organizers and participants.  [Ham82].
Let M be any smooth manifold that is of finite-type. The Abelian groups in the lower horizontal sequence in the diagram (2.11) are finitely generated discrete groups, hence we shall equip them with the discrete topology and therewith obtain zero-dimensional Abelian Fréchet-Lie groups. All arrows in the lower horizontal sequence in (2.11) then become morphisms of Abelian Fréchet-Lie groups. Next, we consider the upper horizontal sequence in (2.11). We endow the R-vector space of differential p-forms Ω p (M ) with the natural C ∞ -topology, i.e. the topology of uniform convergence together with all derivatives on any compact set K ⊂ M . An elegant way to describe the C ∞ -topology is by choosing an auxiliary Riemannian metric g on M and a countable compact exhaustion K 0 ⊂ K 1 ⊂ · · · ⊂ K n ⊂ K n+1 ⊂ · · · ⊂ M , with n ∈ N. We define the family of semi-norms ω l,n := max for all l, n ∈ N and ω ∈ Ω p (M ), where D j : Ω p (M ) → Γ ∞ M, p T * M ⊗ j T * M is the symmetrized covariant derivative corresponding to the Riemannian metric g and | · | is the fibre metric on p T * M ⊗ j T * M induced by g. The C ∞ -topology on Ω p (M ) is the Fréchet topology induced by the family of semi-norms · l,n with l, n ∈ N. It is easy to check that this topology does not depend on the choice of Riemannian metric g and compact exhaustion K n , as for different choices of g and K n the corresponding semi-norms can be estimated against each other.
The subspace of exact forms dΩ k−1 (M ) ⊆ Ω k (M ) is a closed subspace in the C ∞ -topology on Ω k (M ), hence dΩ k−1 (M ) is a Fréchet space in its own right. Forgetting the multiplication by scalars, the Abelian group dΩ k−1 (M ) (with respect to +) in the upper right corner of (2.11) is an Abelian Fréchet-Lie group. Let us now describe the Abelian Fréchet-Lie group structure on Ω k−1 (M )/Ω k−1 Z (M ). For us it will be convenient to provide an explicit description by using charts. As model space we shall take the Fréchet space

A.1 Isomorphism types
We shall now identify the isomorphism types of the Fréchet-Lie groups Ω k−1 (M )/Ω k−1 Z (M ) and H k (M ; Z) by splitting the rows in the diagram (2.11). The lower row splits since H k free (M ; Z) is a free Abelian group and all groups in the lower row carry the discrete topology. By construction, all rows in (2.11) are central extensions of Abelian Fréchet-Lie groups. In particular, they define principal bundles over the groups in the right column. In the following we denote the k-th Betti number of M by b k with k ∈ N; then all b k < ∞ by the assumption that M is of finite-type. Thus we obtain a continuous projection p : Ω k−1 d (M ) → span R ω 1 , . . . , ω b k−1 with kernel dΩ k−2 (M ). Denote by pr j : span R ω 1 , . . . , ω b k−1 → Rω j the projection to the j-th component. Then the continuous linear functionals p j := pr j • p : Ω k−1 d (M ) → Rω j , with j ∈ {1, . . . , b k−1 }, have continuous extensions to Ω k−1 (M ), and so does their direct sum p = p 1 ⊕ · · · ⊕ p b k−1 : Ω k−1 d (M ) → span R ω 1 , . . . , ω b k−1 . Then put F k−1 (M ) := ker(p) to obtain a decomposition of Ω k−1 (M ) as claimed. By construction, the exterior differential induces a continuous isomorphism of Fréchet spaces d : For the middle row in the diagram (2.11), since the connected components of Ω k Z (M ) are contractible, the corresponding principal H k−1 (M ; T)-bundle curv : H k (M ; Z) → Ω k Z (M ) is topologically trivial. We can also split the middle exact sequence in the diagram (2.11) as a central extension: Choose differential forms ϑ 1 , . . . , ϑ b k ∈ Ω k d (M ) whose de Rham classes form a Z-module basis of H k free (M ; Z); this yields a splitting of Ω k Z (M ) analogous to the one in (A.11b). Thus we may write any form µ ∈ Ω k Z (M ) as µ = b k i=1 a i ϑ i + dν, where a i ∈ Z and ν ∈ Ω k−1 (M ). Now choose elements h ϑ i ∈ H k (M ; Z) with curvature curv(h ϑ i ) = ϑ i , for all i = 1, . . . , b k . By the splitting (A.13) we may choose a Fréchet-Lie group homomorphism σ : dΩ k−1 (M ) → Ω k−1 (M )/Ω k−1 Z (M ) such that d • σ = id dΩ k−1 (M ) . Now we define a splitting of the middle row in the diagram (2.11) by setting (A.14) By construction, σ is a homomorphism of Abelian Fréchet-Lie groups, i.e. it is a smooth group homomorphism. Moreover, for any form µ = b k i=1 a i ϑ i + dν ∈ Ω k Z (M ) we have curv(σ(µ)) = b k i=1 a i curv(h ϑ i ) + dν = µ. Thus σ is a splitting of the middle horizontal sequence of Abelian Fréchet-Lie groups in the diagram (2.11), and we have obtained a (noncanonical) decomposition