Abelian duality on globally hyperbolic spacetimes

We study generalized electric/magnetic duality in Abelian gauge theory by combining techniques from locally covariant quantum field theory and Cheeger-Simons differential cohomology on the category of globally hyperbolic Lorentzian manifolds. Our approach generalizes previous treatments using the Hamiltonian formalism in a manifestly covariant way and without the assumption of compact Cauchy surfaces. We construct semi-classical configuration spaces and corresponding presymplectic Abelian groups of observables, which are quantized by the CCR-functor to the category of $C^*$-algebras. We demonstrate explicitly how duality is implemented as a natural isomorphism between quantum field theories. We apply this formalism to develop a fully covariant quantum theory of self-dual fields.


Introduction and summary
Dualities in string theory have served as a rich source of conjectural relations between seemingly disparate situations in mathematics and physics, particularly in some approaches to quantum field theory. Heuristically, a 'duality' is an equivalence between two descriptions of the same quantum theory in different classical terms, and it typically involves an interchange of classical and quantum data. The prototypical example is electric/magnetic duality of Maxwell theory on a four-manifold M : Magnetic flux is discretized at the classical level by virtue of the fact that it originates as the curvature of a line bundle on M , whereas electric flux discretization is a quantum effect arising via Dirac charge quantization. The example of electric/magnetic duality in Maxwell theory has a generalization to any spacetime dimensionality, of relevance to the study of fluxes in string theory, which we may collectively refer to as 'Abelian duality'. The configuration spaces of these (generalized) Abelian gauge theories are mathematically modeled by suitable (generalized) differential cohomology groups, see e.g. [Fre00,Sza12] for reviews.
In this paper we will describe a new perspective on Abelian duality by combining methods from Cheeger-Simons differential cohomology and locally covariant quantum field theory; this connection between Abelian gauge theory and differential cohomology was originally pursued by [BSS14]. The quantization of Abelian gauge theories was described from a Hamiltonian perspective by [FMS07a,FMS07b], where the representation theory of Heisenberg groups was used to define the quantum Hilbert space of an Abelian gauge theory in a manifestly duality invariant way. In the present work we shall instead build the semi-classical configuration space for dual gauge field configurations in a fully covariant fashion, which agrees with that proposed by [FMS07a,FMS07b] upon fixing a Cauchy surface Σ in a globally hyperbolic spacetime M , but which is manifestly independent of the choice of Σ. Following the usual ideas of algebraic quantum field theory, we construct not a quantum Hilbert space of states but rather a C * -algebra of quantum observables; the requisite natural presymplectic structure also agrees with that of [FMS07a,FMS07b] upon fixing a Cauchy surface Σ, but is again independent of the choice of Σ. Our approach thereby lends a new perspective on the phenomenon of Abelian duality, and it enables a rigorous (functorial) definition of quantum duality as a natural isomorphism between quantum field theory functors. An alternative rigorous perspective on Abelian duality has been recently proposed by [Ell14] using the factorization algebra approach to (Euclidean) quantum field theory. We do not yet understand how to describe the full duality groups, i.e. the analogues of the SL(2, Z) S-duality group of Maxwell theory, as this in principle requires a detailed understanding of the automorphism groups of our quantum field theory functors [Few13], which is beyond the scope of the present paper.
Our approach also gives a novel and elegant formulation of the quantum theory of self-dual fields, which is an important ingredient in the formulation of string theory and supergravity: In two dimensions the self-dual gauge field is a worldsheet periodic chiral scalar field in heterotic string theory whose quantum Hilbert space carries representations of the usual (affine) Heisenberg algebra; in six dimensions the self-dual field is an Abelian gerbe connection which lives on the worldvolume of M5-branes and NS5-branes, and in the evasive superconformal (2, 0) theory whose quantum Hilbert space should similarly carry irreducible representations of the corresponding Heisenberg group; in ten dimensions the self-dual field is the Ramond-Ramond four-form potential of Type IIB supergravity. The two generic issues associated with the formulation of the self-dual field theory are: (a) The lack of covariant local Lagrangian formulation of the theory (without certain choices, cf. [BM06]); and: (b) The reconciliation of the self-duality equation with Dirac quantization requires the simultaneous discretization of both electric and magnetic fluxes in the same semi-classical theory. Our quantization of Abelian gauge theories at the level of algebras of quantum observables eludes both of these problems. In particular, the noncommutativity of torsion fluxes observed by [FMS07a,FMS07b] is also straightforwardly evident in our approach. As in [FMS07a,FMS07b], our quantization of the self-dual field does not follow from the approach developed in the rest of this paper. Other Abelian self-dual gauge theories can be analyzed starting from generalized differential cohomology theories fulfilling a suitable self-duality property, e.g. differential K-theory, see [FMS07a,FMS07b] for the Hamiltonian point of view. An approach closer to the one pursued in the present paper is possible also in these cases provided one has suitable control on the properties of the relevant generalized differential cohomology theory.
In addition to being cast in a manifestly covariant framework, another improvement on the development of [FMS07a,FMS07b] is that our approach does not require the spacetime to admit compact Cauchy surfaces. Our main technical achievement is the development of a suitable theory of Cheeger-Simons differential characters with compact support and Pontryagin duality, in a manner which does not destroy the Abelian duality. As the mathematical details of this theory are somewhat involved and of independent interest, they have been delegated to a companion paper [BBSS15] to which we frequently refer. The present paper focuses instead on the aspects of interest in physics.
The outline of the remainder of this paper is as follows. In Section 2 we introduce and analyze the semi-classical configuration spaces of dual gauge fields in the language of differential cohomology; our main result is the identification of this space with the space of solutions of a well-posed Cauchy problem which agrees with the description of [FMS07a,FMS07b], but in a manifestly covariant fashion and without the assumption of compactness of Cauchy surfaces. In Section 3 we analogously study a suitable space of dual gauge field configurations of spacelike compact support, and show in Section 4 that it is isomorphic to a suitable Abelian group of observables defined in the spirit of smooth Pontryagin duality as in [BSS14]. In Section 5 we consider the quantization of the semi-classical gauge theories and the extent to which they satisfy the axioms of locally covariant quantum field theory [BFV03]; we show that, just as in [BSS14], our quantum field theory functors satisfy the causality and time-slice axioms but violate the locality axiom. 1 In Section 6 we show that dualities extend to the quantum field theories thus defined. In Section 7 we apply our formalism to give a proper covariant formulation of the quantum field theory of a self-dual field. An appendix at the end of the paper provides some technical details of constructions which are used in the main text.

Dual gauge fields
In this section we describe and analyze the configuration spaces of the (higher) gauge theories that will be of interest in this paper. Their main physical feature is a discretization of both electric and magnetic fluxes, which is motivated by Dirac charge quantization. To simplify notation, we normalize both electric and magnetic fluxes so that they are quantized in the same integer lattice Z ⊂ R. Because Dirac charge quantization arises as a quantum effect (i.e. it depends on Planck's constant , which in our conventions is equal to 1), we shall use the attribution "semi-classical" for the gauge field configurations introduced below. In this paper all manifolds are implicitly assumed to be smooth, connected, oriented and of finite type, i.e. they admit a finite good cover.

Semi-classical configuration space
Let M be a manifold. The integer cohomology group H k (M ; Z) of degree k is an Abelian group which has a (non-canonical) splitting H k (M ; Z) ≃ H k free (M ; Z) ⊕ H k tor (M ; Z) into free and torsion subgroups, respectively. Let Ω k Z (M ) ⊂ Ω k (M ) denote the closed differential kforms on M with integer periods. Below we recall the definition of Cheeger-Simons differential characters [CS85].
Definition 2.1. A degree k Cheeger-Simons differential character on a manifold M is a group homomorphism h : Z k−1 (M ) → T from the group Z k−1 (M ) of k − 1-cycles on M to the circle group T := R/Z for which there exists a differential form ω h ∈ Ω k (M ) such that where ∂γ denotes the boundary of the k-chain γ. The Abelian group of Cheeger-Simons differential characters is denoted byĤ k (M ; Z).
For a modern perspective on differential cohomology which includes the Cheeger-Simons model see [SS08,BB14]. We use the degree conventions of [BB14] in which the curvature of a differential character inĤ k (M ; Z) is a k-form. The assignment ofĤ k (M ; Z) to each manifold M is a contravariant functorĤ from the category Man of manifolds to the category Ab of Abelian groups. For notational convenience, we simply denote by f * the group homomorphismĤ k (f ; for any smooth map f : M → M ′ . The functor (2.2) comes together with four natural transformations which are given by the curvature map curv :Ĥ k (−; Z) ⇒ Ω k Z (−), the characteristic class map char :Ĥ k (−; Z) ⇒ H k (−; Z), the inclusion of topologically trivial fields 1 The violation of locality is due to topological properties of the spacetime M and owes to the fact that differential cohomology constructs the pertinent configuration spaces as gauge orbit spaces. As a matter of fact, all approaches to gauge theory in the context of general local covariance [BFV03] exhibit at least some remnant of the failure of locality, see [BSS14,BDHS14,BDS14a,DL12,FL14,DS13,SDH14]. There are indications that the tension between locality and gauge theory can be solved by means of homotopical techniques (in the context of model categories), see [BSS15] for the first steps towards this goal. ι : Ω k−1 (−)/Ω k−1 Z (−) ⇒Ĥ k (−; Z) and the inclusion of flat fields κ : H k−1 (−; T) ⇒Ĥ k (−; Z), where T = R/Z is the circle group. The (functorial) diagram of Abelian groups is a commutative diagram whose rows and columns are short exact sequences.
In the remainder of this paper we shall take M to be a time-oriented m-dimensional globally hyperbolic Lorentzian manifold, which we regard as 'spacetime'; for a thorough discussion of Lorentzian geometry including global hyperbolicity see e.g. [BEE96,O'N83], while a brief overview can be found in e.g. [BGP07, Section 1.3]. The semi-classical configuration space C k (M ; Z) of interest to us is obtained as the pullback By definition, any element (h,h) ∈ C k (M ; Z) ⊆Ĥ k (M ; Z) ×Ĥ m−k (M ; Z) has the property that the curvature of h is the Hodge dual of the curvature ofh, i.e. curv h = * curvh. We may interpret this condition as being responsible for the quantization of electric fluxes: the de Rham cohomology class of the Hodge dual curvature * curv h is also an element in H m−k free (M ; Z) and hence electric fluxes are quantized in the same lattice Z ⊂ R as magnetic fluxes. In a similar fashion, we introduce the semi-classical topologically trivial fields T k (M ; Z) as the pullback (2.5) To simplify notation we will adopt the following useful convention: For any graded Abelian group A ♯ = k∈Z A k , we introduce (2.6) Using (2.3) we introduce a new commutative diagram of Abelian groups with exact rows and columns, whose central object is the semi-classical configuration space C k (M ; Z).
Theorem 2.2. Consider the two group homomorphisms Then the diagram of Abelian groups is a commutative diagram whose rows and columns are short exact sequences.
Proof. Commutativity of this diagram follows by construction. Hence we focus on proving that the rows and columns are exact. The bottom row and the left column are exact because they are Cartesian products of exact sequences. Injectivity of ι × ι, κ × κ andκ ×κ is immediate by (2.3).
We still have to check that the first two rows and the last two columns are exact at their middle objects. This is a straightforward consequence of the exactness of the corresponding rows and columns in (2.3).
Remark 2.3. To better motivate the semi-classical configuration space C k (M ; Z) we establish below its relation with Maxwell theory. For this purpose we consider the case m = 4 and k = 2. The usual Maxwell equations (without external sources) for the Faraday tensor F ∈ Ω 2 (M ) are dF = 0 and d * F = 0. These equations are invariant under electric-magnetic duality, i.e. under the exchange of F and * F . The standard approach to gauge theory consists in the replacement of F with the curvature of (the isomorphism class of) a circle bundle with connection (equivalently, a differential cohomology class in degree 2). In this framework, however, * F does not have any geometric interpretation, hence the original electric-magnetic duality of Maxwell theory is lost passing to gauge theory. Nevertheless, one can present Maxwell equations in an equivalent way, which is however better suited for a gauge theoretic extension preserving electric-magnetic duality: Interpreting both F andF as the curvatures of circle bundles with connections, the semiclassical configuration space C 2 (M ; Z) is obtained and the original electric-magnetic duality of Maxwell theory is lifted to C 2 (M ; Z), see Section 6 for the situation in arbitrary spacetime dimension and degree. Notice that the semi-classical configuration space has the same local "degrees of freedom" as Maxwell theory. In fact, on a contractible spacetime C 2 (M ; Z) reduces to the top-right corner in diagram (2.8). Since exact and closed forms are the same on a contractible manifold, Maxwell theory is recovered. In conclusion, the semi-classical configuration space C 2 (M ; Z) is a gauge theoretic extension of Maxwell theory that carries the same local information, however preserving electric-magnetic duality by matching the relevant topological (as opposed to local) data in a suitable way. As a by-product, any configuration (h,h) ∈ C 2 (M ; Z) realizes the discretization of magnetic and electric fluxes, which arise as the characteristic classes char h, charh ∈ H 2 (M ; Z). This argument can be made general for higher gauge theories in arbitrary spacetime dimension.
Remark 2.4. The semi-classical configuration space is a contravariant functor from the category Loc m of time-oriented m-dimensional globally hyperbolic Lorentzian manifolds with causal embeddings 3 as morphisms to the category Ab of Abelian groups. For notational convenience, we simply denote by f * the group homomorphism

Cauchy problem
We will now show that the semi-classical configuration space C k (M ; Z) is the space of solutions of a well-posed Cauchy problem. Let us start by recalling a well-known result for the Cauchy 3 A causal embedding f : M → M ′ between time-oriented m-dimensional globally hyperbolic Lorentzian manifolds is an orientation and time-orientation preserving isometric embedding, whose image is open and causally compatible, i.e.
) for all p ∈ M ; here J ± M (p) denotes the causal future/past of p ∈ M consisting of all points of M which can be reached by a future/past-directed smooth causal curve stemming from p, see [BGP07].
problem of the Faraday tensor, see e.g. [DL12,FL14] and also [BF09, Chapter 3, Corollary 5] for details on how to treat initial data of not necessarily compact support. For the related Cauchy problem of the gauge potential see [SDH14]. Throughout this paper Σ will denote a smooth spacelike Cauchy surface of M with embedding ι Σ : Σ → M into M .
whose support is contained in the causal future and past of the support of the initial data, i.e. supp F ⊆ J(supp B ∪ suppB).
We consider also the similar well-posed initial value problem forF ∈ Ω m−k (M ) given by where the initial data are also specified by Given now any initial data (B,B) ∈ Ω k,m−k d (Σ), let us consider the corresponding unique solutions F andF of the Cauchy problems (2.11) and (2.12). This implies that F − * F solves the Cauchy problem (2.11) with vanishing initial data, and therefore F = * F . We further show that, given initial data (B,B) ∈ Ω k,m−k Z (Σ) with integral periods, the corresponding solution F of the Cauchy problem (2.11) is such that both F and * −1 F have integral periods. For this, using the results of Lemma A.1 (i) we can express each k-cycle γ ∈ Z k (M ) as γ = ι Σ * π Σ * γ + ∂h Σ γ, and hence (2.13) (2.14) . Summing up, we obtain Corollary 2.6. The embedding ι Σ : Σ → M of Σ into M induces an isomorphism of Abelian groups Let us consider the central row of the diagram (2.8). Taking into account also naturality of κ and curv, one finds that the diagram of Abelian groups commutes and its rows are short exact sequences. Using also Lemma A.1 (ii), Corollary 2.6 and the five lemma, we obtain Theorem 2.7. The embedding ι Σ : Σ → M induces an isomorphism of Abelian groups We can interpret the result of Theorem 2.7 as establishing the well-posedness of the initial value problem for (h,h) ∈Ĥ k,m−k (M ; Z) given by . It follows that the semi-classical configuration space C k (M ; Z) arises as the space of solutions of this Cauchy problem.
Remark 2.8. If M has compact Cauchy surfaces Σ, we can easily endow C k (M ; Z) with the structure of a presymplectic Abelian group induced by the ring structure · on differential characters, see [CS85,SS08,BB14]. For this, we define the circle-valued presymplectic structure denotes the fundamental class of Σ. Using compatibility between the ring structure on differential characters and the natural transformations ι, κ, curv and char, one can show that σ is in fact independent of the choice of Σ. Fixing any Cauchy surface Σ and using the isomorphism given in Theorem 2.7, the presymplectic structure (2.19) can be induced to initial data and thereby agrees with the one constructed by [FMS07b,FMS07a] from a Hamiltonian perspective. However, in contrast to [FMS07b,FMS07a] our construction does not depend on the choice of a Cauchy surface, i.e. it is generally covariant. As we show in Section 4, the assumption of compactness of the Cauchy surfaces can be dropped, provided that one introduces a suitable support restriction on the semi-classical gauge fields.

Dual gauge fields with spacelike compact support
In this section we introduce and analyze a suitable Abelian group C k sc (M ; Z) of semi-classical gauge fields of spacelike compact support. Similarly to the case of the usual quantum field theories on curved spacetimes, such as Klein-Gordon theory, the role played by C k sc (M ; Z) will be dual to that of the semi-classical configuration space C k (M ; Z); in fact, we shall show in Section 4 that elements in C k sc (M ; Z) define functionals (i.e. classical observables) on C k (M ; Z) which are group characters C k (M ; Z) → T. This dual role of the semi-classical gauge fields of spacelike compact support will be reflected mathematically in the fact that C k is a very subtle point because, in contrast to the standard examples like Klein-Gordon theory, the Abelian group C k sc (M ; Z) cannot be presented as a subgroup of C k (M ; Z), see Remark 3.1 below. We give a definition of C k sc (M ; Z) in terms of relative differential cohomology and frequently refer to the companion paper [BBSS15] for further technical details.

Semi-classical configuration space
Let K ⊆ M be a compact subset. In analogy to (2.4), we define the Abelian group C k (M, M \ J(K); Z) of semi-classical gauge fields on M relative to M \ J(K) as the pullback whereĤ p (M, M \ J(K); Z) denote the relative differential cohomology groups and Ω k (M, M \ J(K)) denotes the group of relative differential forms, see [BB14,BBSS15] for the definitions and our conventions. We shall make frequent use of the short exact sequence for relative semi-classical gauge fields, which immediately follows from [BB14, Part II, Section 3.3] and [BBSS15, Theorem 3.2] by imitating the proof of Theorem 2.2.
Remark 3.1. One may heuristically think of semi-classical gauge fields on M relative to M \ J(K) as fields on M which "vanish" outside of the closed light-cone J(K) of K. However, strictly speaking this interpretation is not correct: There is a group homomorphism which is induced by the group homomorphisms (denoted with abuse of notation by the same symbols) I :Ĥ p (M, M \ J(K); Z) →Ĥ p (M, Z) that restrict relative differential characters from relative cycles to cycles by precomposing them with the ho- We define the Abelian group which is by construction a compact subset of Σ. This observation provides the isomorphism Similarly to Remark 3.1, there is a group homomorphism (denoted with abuse of notation by the same symbol) which is however in general not injective, see [BBSS15,Remark 4.4] and Corollary 3.5 below.
Hence semi-classical gauge fields of spacelike compact support cannot in general be faithfully represented as elements in the semi-classical configuration space C k (M ; Z).

Cauchy problem
Consider any compact subset K ⊆ Σ. Taking into account the support property of the Cauchy problem considered in Theorem 2.5 and applying arguments similar to those in Section 2.2 to the relative case, in particular (2.13) and (2.14) (see also Lemma A.2 (i)), one concludes that, given initial data (B,B) ∈ Ω k,m−k Z (Σ, Σ \ K), the Cauchy problem (2.11) has a unique solution . This observation leads us to the relative version of Corollary 2.6.
Using (3.2) and [BBSS15, Theorem 3.2], and the fact that relative differential cohomology is a functor (in a suitable sense, see [BBSS15, Section 3.1]), we conclude that the diagram of Abelian groups commutes and its rows are short exact sequences. Using also Lemma A.2 (ii), Corollary 3.3 and the five lemma, we obtain the relative version of Theorem 2.7.
Theorem 3.4. The embedding ι Σ : Σ → M induces an isomorphism of Abelian groups Taking the colimit of (3.7) over the directed set K Σ of compact subsets of Σ and recalling Remark 3.2 we find that the diagram of Abelian groups commutes, its rows are short exact sequences and its vertical arrows are isomorphisms. The subscript c denotes compact support and the various groups of this diagram are defined by these colimits. 5 This shows that C k sc (M ; Z) is the space of solutions of the Cauchy problem commutes. Here we also use the notation f * for the group homomorphisms H k−1,m−k−1 sc (f ; T) and Ω k sc,Z ∩ * Ω m−k sc,Z (f ).

Semi-classical observables
We shall begin by imposing a suitable regularity condition on the Abelian group of group characters C k (M ; Z) ⋆ of the semi-classical configuration space. We now prove that Definition 4.1 does not depend on the choice of Cauchy surface Σ used to evaluate the integral (4.1). For this, notice that ω = * ω ∈ Ω k sc,Z ∩ * Ω m−k sc,Z (M ) implies ω = 0 and ω = 0, where = δ d + d δ is the d'Alembert operator. By [BGP07,Bär15] there exists β ∈ Ω m−k c (M ) such thatω = Gβ, where G = G + − G − is the causal propagator and G ± are the retarded/advanced Green's operators of . We further have ω = * ω = G * β. Because of dω = 0 and dω = 0, there exist α ∈ Ω k+1 c (M ) andα ∈ Ω m−k+1 c (M ) such that d * β = α and dβ = α. Using these observations, and realizing Σ as the boundary of J − (Σ) ⊆ M and also as the boundary of J + (Σ) ⊆ M (with opposite orientation), we can rewrite (4.1) as where we have also used dA = * dÃ. It then follows that (4.1) is independent of the choice of Cauchy surface because (4.2) shows that it can be written as an integral over spacetime M . . This pushout may be realized explicitly as the quotient . (4.4) One can show that the Abelian group C k (M ; Z) ⋆ ∞ is isomorphic to the Abelian group O k (M ; Z) of semi-classical observables given in Definition 4.1. As we do not need this isomorphism in this paper, we refrain from writing it out explicitly. Let us just point out that the elements of the smooth Pontryagin dual are in particular continuous group characters. In fact, on account of [BSS14, Appendix A], all differential cohomology groups on a manifold of finite type are Fréchet-Lie groups that are (non-canonically) isomorphic to the Cartesian product of a torus, a torsion group, a discrete lattice in a Euclidean space (all finite dimensional) and a Fréchet vector space of differential forms. This observation allows one to conclude that the elements of the smooth Pontryagin dual are continuous group characters with respect to the Fréchet topology mentioned above.  ) and ω = * ω ∈ Ω k sc,Z ∩ * Ω m−k sc,Z (M ) be as in Definition 4.1. Exploiting the Cauchy problem described by Corollary 3.3, we can easily push forward ω and ω to f * ω = * f * ω ∈ Ω k sc,Z ∩ * Ω m−k sc,Z (M ′ ) by pushing forward the initial data from a Cauchy surface Σ ⊆ M to a suitable Cauchy surface Σ ′ ⊆ M ′ . 6 By construction, we have which shows that f * ϕ ∈ O k (M ′ ; Z) as required.

Observables from spacelike compact gauge fields
We shall now show that the Abelian group By [BBSS15, Section 5.2], for any smooth spacelike Cauchy surface Σ of M there is a T-valued pairing ·, · c :Ĥ m−p (Σ; Z) ×Ĥ p c (Σ; Z) → T between differential cohomology and compactly supported differential cohomology. Using the isomorphisms given in Theorem 2.7 and Corollary 3.5, we define a T-valued pairing between C k (M ; Z) and C k sc (M ; Z) by In Lemma A.4 we show that this pairing does not depend on the choice of Cauchy surface Σ and we prove its naturality in the sense that for any morphism f : M → M ′ in Loc m the diagram of Abelian groups

commutes.
By partial evaluation, the pairing (4.7) allows us to define group characters on C k (M ; Z): The next result in particular allows us to separate points of the semi-classical configuration space C k (M ; Z) by using only such group characters.  Finally, we show that the partial evaluation (4.9) establishes an isomorphism between  Proof. We first have to show that the group character · , (h ′ ,h ′ ) satisfies the regularity condition of Definition 4.1, for any (h ′ ,h ′ ) ∈ C k sc (M ; Z). This follows from [BBSS15, eq. (4.12)]: For any ([A], [Ã]) ∈ T k (M ), we find Moreover, (4.11) is injective due to Proposition 4.3.
To show that (4.11) is surjective, let us take any ϕ ∈ O k (M ; Z) and choose a smooth spacelike Cauchy surface Σ of M . Using the isomorphism established in Theorem 2.7 and recalling Definition 4.1, there exists a unique smooth character . Then the definition of ·, · given in (4.7) implies that ϕ = · , (h ′ ,h ′ ) . It remains to prove that the established isomorphism is natural.

Presymplectic structure
We will introduce a natural T-valued presymplectic structure τ on the Abelian group O k (M ; Z) of semi-classical observables. In this way we obtain a functor (O k (−; Z), τ ) : Loc m → PSAb valued in the category PSAb of presymplectic Abelian groups (with group homomorphisms preserving the presymplectic structures as morphisms). This will be the main input for Section 5, where the quantization of the semi-classical model described by (O k (−; Z), τ ) will be addressed. Then defines a presymplectic structure on O k (M ; Z) whose radical is O(ker I).
Proof. Up to the isomorphism O −1 : O k (M ; Z) → C k sc (M ; Z), the mapping τ is given by Remark 4.6. If M has compact Cauchy surfaces Σ ⊆ M , the group homomorphism I : C k sc (M ; Z) → C k (M ; Z) is the identity. In fact, with Σ compact, J(Σ) = M entails that the diagram whose colimit defines C k sc (M ; Z) has C k (M, M \ J(Σ); Z) = C k (M ; Z) as its terminal object. In particular, τ is actually weakly symplectic for globally hyperbolic Lorentzian manifolds with compact Cauchy surfaces. In this case (4.15) coincides with the (pre)symplectic structure described in Remark 2.8. Our next task is to prove that the presymplectic structure τ introduced in Proposition 4.5 is natural, so that we can interpret (O k (−; Z), τ ) as a functor from Loc m to PSAb.

Locally covariant field theory
and the proof follows from f * 2 • I • f 1 * = 0, see Lemma A.5.   where ⋆ denotes Pontryagin duality. The first isomorphism is from (3.9), the second is presented in [BBSS15, Remark 5.7] and the third simply follows from homotopy invariance of cohomology and M ≃ R × Σ. Hence the counterexamples to injectivity provided in [BSS14, Example 6.9] can be used to prove the present claims. For the case m = 2 and k = 1, see the argument preceding [BSS14, Proposition 6.11].
The next theorem summarizes the results obtained in this section in view of the standard axioms of locally covariant field theory [BFV03]. In particular, we stress that the locality axiom, which requires f  More precisely, the CCR-functor CCR : PSAb → C * Alg from the category of presymplectic Abelian groups to the category of C * -algebras is constructed in detail in [BDHS14, Appendix A]. Composing the functor (O k (−; Z), τ ) : Loc m → PSAb with the CCR-functor, we obtain a functor from Loc m to C * Alg which, according to [BFV03], should be interpreted as a quantum field theory. We can thereby define a family of quantum field theories by setting

Quantum duality
In this section we show that there exist dualities between the quantum field theories defined in (5.2). These dualities will be described at the functorial level and therefore hold true for all spacetimes M in a coherent (natural) way. In order to motivate our definition of duality given below, let us recall that a quantum field theory is a functor A : Loc m → C * Alg from the category of m-dimensional spacetimes to the category of C * -algebras. The collection of all m-dimensional quantum field theories is therefore described by the functor category [Loc m , C * Alg]; objects in this category are functors A : Loc m → C * Alg and morphisms are natural transformations η : A ⇒ A ′ . In physics one calls the functor category [Loc m , C * Alg] the "theory space" of m-dimensional quantum field theories which, being a category, comes with a natural notion of equivalence of theories.
Definition 6.1. A duality between two quantum field theories A, A ′ : Loc m → C * Alg is a natural isomorphism η : A ⇒ A ′ .
We shall now construct explicit dualities between the quantum field theories A k and A m−k given in (5.2), for all m ≥ 2 and k ∈ {1, . . . , m − 1}. Our strategy is to define first the dualities at the level of the semi-classical configuration spaces (2.4), and then lift them to the presymplectic Abelian groups and ultimately to the corresponding quantum field theories. For any object M in Loc m we define a group homomorphism commutes. We next observe that (6.4) preserves the presymplectic structure (4.15): A quick calculation shows that Using also the natural isomorphisms O : C p sc (−; Z) ⇒ O p (−; Z) given in Proposition 4.4, for p = k and p = m−k, we find that ζ ⋆ defines a natural isomorphism (denoted by the same symbol) between functors from Loc m to PSAb.
We can now state the main result of this section.
Theorem 6.2. The C * -algebra homomorphism defines a duality between the two quantum field theories A k , A m−k : Loc m → C * Alg.
Proof. We need to show that η defines a natural isomorphism η : A k ⇒ A m−k . Naturality of η is a direct consequence of naturality of ζ ⋆ and the fact that CCR is a functor, in particular it preserves compositions. As functors preserve isomorphisms it then follows that η is a natural isomorphism.
Corollary 6.3. For m = 2k the duality of Theorem 6.2 becomes a self-duality, i.e. a natural automorphism η : A k ⇒ A k .

Self-dual Abelian gauge theory
In dimension m = 2k it makes sense to demand the self-duality condition curv h = * curv h (7.1) for a differential character h ∈Ĥ k (M ; Z). Applying the Hodge operator * to both sides of (7.1) we obtain * curv h = −(−1) k 2 curv h = −(−1) k 2 * curv h , (7.2) which implies that for k even the only solutions to (7.1) are flat fields h = κ(t), for t ∈ H k−1 (M ; T). In the following we shall focus on the physically much richer and interesting case where k ∈ 2Z ≥0 + 1 is odd.
The Abelian group of solutions to the self-duality equation (7.1) is denoted There is a monomorphism to the semi-classical configuration space introduced in (2.4). Given any smooth spacelike Cauchy surface Σ of M with embedding ι Σ : Σ → M , we compose (7.4) with the isomorphism of Theorem 2.7 and obtain a monomorphism to the Abelian group of semi-classical gauge fields of spacelike compact support introduced in (3.3). Using Corollary 3.5, one easily shows that is an isomorphism, which we may interpret as establishing the well-posedness of the initial value problem (7.7) for h ∈Ĥ k sc (M ; Z) of spacelike compact support and initial datum h Σ ∈Ĥ k c (Σ; Z) of compact support.
Similarly to (4.7), there is a weakly non-degenerate T-valued pairing Analogously to Proposition 4.5 we define a T-valued presymplectic structure Up to the isomorphism O −1 sd : sdO k (M ; Z) → sdC k sc (M ; Z) induced by (7.12), the presymplectic structure reads as (7.14) The radical of σ sd coincides with the kernel of I : sdC k sc (M ; Z) → sdC k (M ; Z), hence the radical of τ sd is O sd (ker I).
Using arguments similar to those of Section 4, one can show that the presymplectic Abelian groups (sdO k (M ; Z), τ sd ) for the self-dual field theory are functorial, i.e. we have constructed a functor (sdO k (−; Z), τ sd ) : Loc 2k −→ PSAb . (7.15) Composing with the CCR-functor from Section 5 we obtain quantum field theories for all k ∈ 2Z ≥0 + 1, which quantize the self-duality equation (7.1). Using similar arguments as those of Section 4.4, one can show that these quantum field theories satisfy the same properties as those listed in Theorem 5.1.
Theorem 7.1. The functor sdA k : Loc 2k → C * Alg enjoys the following properties: • Quantum causality axiom: Let M 1 ←− M 2 be a diagram in Loc 2k such that the images of f 1 and f 2 are causally disjoint. Then the subalgebras f 1 * (sdA k (M 1 )) and f 2 * (sdA k (M 2 )) of sdA k (M ) commute. For k = 1 the latter is always the case, while for k ∈ 2Z ≥0 + 3 there is at least one morphism in Loc 2k violating injectivity.
Remark 7.2. We address the question how the self-dual quantum field theories sdA k , which quantize the self-duality equation (7.1), are related to the self-dualities of the quantum field theories A k established in Corollary 6.3. Let k ∈ 2Z ≥0 + 1 and consider any object M in Loc 2k . The self-duality (6.4) on C k sc (M ; Z) then reduces to ζ ⋆ (h ′ ,h ′ ) = (h ′ , h ′ ), i.e. it simply interchanges h ′ andh ′ . The Abelian group of invariants under this self-duality is given by the diagonal which by (7.9) is isomorphic to sdC k sc (M ; Z). Restricting the presymplectic structure (4.15) to the invariants C k sc (M ; Z) inv then yields where σ sd is the presymplectic structure on sdC k sc (M ; Z) given in (7.14). Due to the prefactor 2, it follows that (sdC k sc (M ; Z), σ sd ) and (C k sc (M ; Z) inv , σ) are not isomorphic as presymplectic Abelian groups, but only as Abelian groups. Moreover, the C * -algebras sdA k (M ) and A k (M ) inv (i.e. the C * -subalgebra of A k (M ) which is generated by the invariant Weyl symbols W(O(h ′ , h ′ )), for all (h ′ , h ′ ) ∈ C k sc (M ; Z) inv ) are in general not isomorphic. Thus even though the quantum field theories sdA k : Loc 2k → C * Alg and A k (−) inv : Loc 2k → C * Alg are similar, they are strictly speaking not isomorphic. In particular, due to effects which are caused by Z 2torsion elements in the cohomology groups H k (M ; Z), the latter theory typically has a bigger center than the former theory. An explicit example of this fact is illustrated below.
Example 7.3. Fix any k ∈ 2Z ≥0 + 3 and consider the lens space L = S 2k−3 /Z 2 obtained as the quotient of the 2k−3-sphere S 2k−3 by the antipodal Z 2 -action. Take any object M in Loc 2k which admits a smooth spacelike Cauchy surface Σ diffeomorphic to T 2 × L, where T 2 is the 2-torus. Since the Cauchy surface Σ is compact, the notion of spacelike compact support becomes irrelevant for this spacetime M and in particular the homomorphism I : sdC k sc (M ; Z) → sdC k (M ; Z) reduces to the identity. Using standard results on the homology groups of lens spaces, see e.g. [Hat02, Chapter 2, Example 2.43], and the universal coefficient theorem for cohomology, one shows that H k−1 (L; Z) ≃ Z 2 . Using the Künneth theorem we find that H k (Σ; Z) has a direct summand (Z 2 ) 2 . In particular, there exists t ∈ H k (Σ; Z) such that t = 0, but 2t = 0. Recalling that char is surjective (cf. (2.3)), we find f ∈Ĥ k (Σ; Z) such that char f = t. It follows that there exists A ∈ Ω k−1 (Σ) such that ι[A] = 2f . Introducing h Σ = f − ι[A/2] ∈Ĥ k (Σ; Z), by construction we obtain h Σ = 0 (otherwise t would be trivial) and 2h Σ = 0. Solving the initial value problem (7.7) provides h ∈ sdC k (M ; Z) with h = 0, but 2h = 0. In fact, 2h ∈ sdC k (M ; Z) solves (7.7) with initial datum 2h Σ = 0. Since Σ is compact, the presymplectic structure (7.14) is weakly non-degenerate. In particular, being non-zero, h ∈ sdC k (M ) is not in the radical. Conversely, taking into account also (h, h) ∈ C k (M ; Z) inv , we find σ((h, h), (h ′ , h ′ )) = 2σ sd (h, h ′ ) = σ sd (2h, h ′ ) = 0 for all (h ′ , h ′ ) ∈ C k (M ) inv . This shows that the center of A k (M ) inv is bigger than that of sdA k (M ) for this particular spacetime M .