Linear Yang-Mills theory as a homotopy AQFT

It is observed that the shifted Poisson structure (antibracket) on the solution complex of Klein-Gordon and linear Yang-Mills theory on globally hyperbolic Lorentzian manifolds admits retarded/advanced trivializations (analogs of retarded/advanced Green's operators). Quantization of the associated unshifted Poisson structure determines a unique (up to equivalence) homotopy algebraic quantum field theory (AQFT), i.e. a functor that assigns differential graded $\ast$-algebras of observables and fulfills homotopical analogs of the AQFT axioms. For Klein-Gordon theory the construction is equivalent to the standard one, while for linear Yang-Mills it is richer and reproduces the BRST/BV field content (gauge fields, ghosts and antifields).


Introduction and summary
Because of their outstanding significance in physics and their intricate connection to mathematics, quantum gauge theories continuously attract a high level of attention throughout different fields of research. In the context of algebraic quantum field theory (AQFT) [HK64,BFV03], which is a powerful axiomatic framework for quantum field theory on Lorentzian manifolds, it is a long-standing open problem to identify the characteristic features of quantum gauge theories and their gauge symmetries from a model-independent perspective. To support these more abstract developments, concrete examples of quantum gauge theories were constructed in the context of AQFT. Most of these studies focused on the case of Yang-Mills theory with structure group R or U (1), see e.g. [SDH14,BDS14,BDHS14,FL16,Ben16,BSS16], but there also exist similar developments for e.g. linearized gravity [FH13,BDM14,Kha16,Kha18] and linearized supergravity [HS13]. In addition to such non-interacting models, examples of perturbatively interacting quantum gauge theories were constructed in [Hol08,FR12,FR13,TZ18] by means of an appropriate adaption of the BRST/BV formalism to AQFT.
One of the main conceptual insights of these studies was the observation that quantum gauge theories, when formulated traditionally in terms of gauge-invariant on-shell observable algebras, are in conflict with crucial axioms of AQFT. The first observation [DL12] was that quantum gauge theories may violate the isotony axiom of AQFT, which demands that the push-forward A(f ) : A(M ) → A(N ) of observables along every spacetime embedding f : M → N is an injective map. It was later understood that the violation of isotony is due to topological charges in quantum gauge theories, e.g. electric and magnetic fluxes in Abelian Yang-Mills theory, and hence it is a feature that is expected on physical grounds, see e.g. [SDH14,BDS14,BDHS14,Ben16,BSS16,BBSS17] for a detailed explanation. The second observation is more subtle as it is related to local-to-global properties (i.e. descent) of AQFTs. Within the traditional formulation in terms of gauge-invariant on-shell observable algebras, quantum gauge theories have very poor local-to-global properties as witnessed for example by the observation in [DL12,FL16] that Fredenhagen's universal algebra (which is a certain local-to-global construction) for Abelian Yang-Mills theory fails to encode crucial gauge theoretic features such as Dirac's charge quantization and Aharonov-Bohm phases. It was later understood and emphasized in [BSS15] that the failure of (too naive versions of) local-to-global constructions is due to higher categorical structures in classical and quantum gauge theories, which are neglected (i.e. truncated) when working in a traditional AQFT setting that is based on gauge-invariant on-shell observables.
Our approach towards resolving this conflict at the interface of AQFT and gauge theory is the recent homotopical AQFT program [BSS15, BS17, BSS18, BSW17, BSW19a, BSW19b, BS18], whose aim is to refine the foundations of AQFT by introducing new concepts from higher category theory. We refer to [BS19] for a recent summary and state-of-the-art review of this approach. Informally speaking, the main difference between a homotopy AQFT and an ordinary AQFT is that it assigns to each spacetime a higher categorical algebra in contrast to an ordinary *algebra of observables, such that suitable homotopy coherent analogs of the AQFT axioms hold true. These statements can be made precise by using techniques from operad theory [BSW17,BSW19a,BSW19b]. Such higher observable algebras should be understood as quantizations of function algebras on higher categorical spaces called (derived) stacks, which are crucial for the description of field and solution spaces in a gauge theory, see e.g. [Sch13] and [BS19] for an introduction and also [BSS18] for a concrete description of the Yang-Mills stack. In the context of linear and perturbative quantum gauge theory, the higher field and solution spaces may be described by chain complexes of vector spaces and the higher quantum observable algebras by differential graded * -algebras. A more physical approach to such higher categorical structures is given by the BRST/BV formalism, which has already found many interesting applications in perturbative AQFT, see e.g. [Hol08,FR12,FR13,TZ18].
One of the most pressing current issues of the homotopical AQFT program is that there is up to now no fully worked out physical example of a quantum gauge theory in this framework.
(Various oversimplified toy-models appeared previously in e.g. [BS17,BSW19b,BS18].) It is the aim of the present paper to address this issue by constructing a first proper example of a homotopy AQFT, namely linear quantum Yang-Mills theory with structure group R on globally hyperbolic Lorentzian manifolds. Let us emphasize that, even though linear Yang-Mills theory is clearly one of the simplest examples of a gauge theory, its construction as a homotopy AQFT is far from trivial because one has to work consistently within a higher categorical context.
A central role in our construction is played by (a linear analog of) the derived critical locus of the linear Yang-Mills action functional, which yields a chain complex that encodes very refined information about the solutions to the linear Yang-Mills equation. By general results of derived algebraic geometry [PTVV13,CPTVV17,Pri18], this chain complex carries a canonical shifted Poisson structure, which is the crucial ingredient in the factorization algebra approach to quantum field theory by Costello and Gwilliam [CG17]. One of our main observations in this paper is that this shifted Poisson structure is trivial in homology due to the geometry of globally hyperbolic Lorentzian manifolds and that it can be trivialized by two different kinds of homotopies that play a similar role as retarded/advanced Green's operators in ordinary field theory. Taking the difference between a retarded and an advanced trivialization allows us to define an unshifted Poisson structure and hence to study the canonical quantization of linear Yang-Mills theory. One of the technical challenges that we address in this paper is a homotopical analysis of the construction sketched above, which is required to ensure that it is meaningful within our higher categorical context, i.e. compatible with quasi-isomorphisms of chain complexes and chain homotopies between Poisson structures. For this we shall use techniques from both model category theory [Hov99,Hir03] and homotopical category theory [DHKS04,Rie14]. In order to make the bulk of this paper accessible to a broader audience, we limit our use of such homotopical techniques to the bare minimum that is required to ensure consistency of our results.
Let us now explain in more detail our constructions and results by outlining the content of the present paper: In Section 2 we recall some preliminary results concerning retarded/advanced Green's operators for Green hyperbolic operators on globally hyperbolic Lorentzian manifolds and concerning chain complexes of vector spaces. These techniques will be frequently used throughout the whole paper. In Section 3 we introduce a flexible concept of field complexes for linear gauge theories and compute the solution complexes corresponding to a quadratic action functional via a linear analog of the derived critical locus construction. We apply these techniques to two explicit examples, given by Klein-Gordon and linear Yang-Mills theory on globally hyperbolic Lorentzian manifolds, and explain how they relate to the BRST/BV formalism from physics. In particular, the derived critical locus construction produces the field content of the BRST/BV formalism, i.e. fields, ghosts and antifields, together with the relevant differentials.
In Section 4 we describe and analyze the shifted Poisson structure on the solution complex that exists canonically due to its construction as a derived critical locus. (In the terminology of the BRST/BV formalism, this is called the antibracket.) Our main novel observation is that, for Klein-Gordon and linear Yang-Mills theory on globally hyperbolic Lorentzian manifolds, this shifted Poisson structure is trivial in homology and that it can be trivialized by two distinct types of homotopies that play a similar role to retarded/advanced Green's operators in ordinary field theory. We formalize this insight by introducing an abstract concept of retarded/advanced trivializations (Definition 4.4). We prove that these trivializations exist for our two running examples and that they are unique in an appropriate sense: For Klein-Gordon theory they are unique, while for linear Yang-Mills theory they are not unique in a strict sense but rather unique up to chain homotopies, which is an appropriate and expected relaxation within our higher categorical context of the uniqueness result for retarded/advanced Green's operators in ordinary field theory. Taking the difference between (a compatible pair of) a retarded and an advanced trivialization allows us to define an unshifted Poisson structure (Definition 4.7), which is again unique up to chain homotopies.
In Section 5 we study in detail homotopical properties of the canonical commutation relations (CCR) quantization of unshifted Poisson complexes into differential graded * -algebras. Our main result in this section is Proposition 5.3, which proves that CCR quantization is compatible with weak equivalences of Poisson complexes and also with homotopies of unshifted Poisson structures. This proof requires a rather technical result that is proven in Appendix A. Hence, the examples obtained by our construction in Section 4 can be quantized consistently. We spell out the quantization of Klein-Gordon and linear Yang-Mills theory in this approach explicitly in Examples 5.5 and 5.6.
In Section 6 we investigate functoriality of our constructions and answer affirmatively our initial question whether they define examples of homotopy AQFTs (Definition 6.1). A key ingredient for these studies is an appropriate concept of natural retarded/advanced trivializations (Definition 6.5) and of natural unshifted Poisson structures (Definition 6.8). Our construction in Section 4 determines such natural structures for both Klein-Gordon and linear Yang-Mills theory, see Proposition 6.7. The main result of this paper is Theorem 6.19, which proves that our construction yields a description of Klein-Gordon and linear Yang-Mills theory as homotopy AQFTs. Concerning uniqueness (up to natural weak equivalences) of our construction, we observe that there are subtle differences between Klein-Gordon and linear Yang-Mills theory. While our construction determines Klein-Gordon theory uniquely (up to natural weak equivalences) on the category Loc of all globally hyperbolic Lorentzian manifolds, we can currently only ensure uniqueness (up to natural weak equivalences) of our construction of linear Yang-Mills theory on each slice category Loc/M , for M ∈ Loc. (See Theorem 6.19 and Remarks 6.20 and 6.21 for the details.) In AQFT terminology, this means that, even though we successfully provide a construction of linear Yang-Mills theory as a homotopy AQFT in the locally covariant framework [BFV03], we can currently only ensure that each of its restrictions to a Haag-Kastler style homotopy AQFT on a fixed spacetime M ∈ Loc is determined uniquely (up to natural weak equivalences) by our methods. This potential non-uniqueness of linear quantum Yang-Mills theory in the locally covariant setting is linked to features of the category of globally hyperbolic spacetimes Loc which, in contrast to the slice categories Loc/M , has no terminal object.

Green's operators
We briefly review those aspects of the theory of Green hyperbolic operators on globally hyperbolic Lorentzian manifolds that are required for this work. The reader is referred to [BGP07,Bar15] for the details.
Let M be an oriented and time-oriented globally hyperbolic Lorentzian manifold of dimension m ≥ 2. Let F → M be a finite-rank real vector bundle and denote its vector space of sections by F(M ) = Γ ∞ (M, F ). A linear differential operator P : F(M ) → F(M ) is called Green hyperbolic if it admits retarded and advanced Green's operators G ± : F c (M ) → F(M ), where the subscript c denotes compactly supported sections. Recall that a retarded/advanced Green's operator is by definition a linear map G ± : F c (M ) → F(M ) which satisfies the following properties: It was proven in [Bar15] that retarded/advanced Green's operators are necessarily unique and that they admit unique extensions to sections with past/future compact support, such that the three properties above hold true for all ϕ ∈ F pc/fc (M ). (Recall that s ∈ F(M ) has past/future compact support if there exists a Cauchy surface Σ ⊂ M such that supp(s) ⊆ J ± M (Σ).) The difference G := G + − G − : F c (M ) → F(M ) of the retarded and advanced Green's operator (on compactly supported sections) is often called the causal propagator. From the properties of G ± it follows that is an exact sequence, where the subscript sc denotes sections of spacelike compact support. (Recall that s ∈ F(M ) has spacelike compact support if there exists a compact subset K ⊆ M such that supp(s) ⊆ J + M (K) ∪ J − M (K).) In particular, this implies that P G = 0 = GP . For every vector bundle F → M that is endowed with a fiber metric h one can define the integration pairing for all s, s ′ ∈ F(M ) with compactly overlapping support. Let us consider two such vector bundles F 1 → M and F 2 → M with fiber metrics and a linear differential operator Q : There exists a formal adjoint differential operator Q * : for all ω, ζ ∈ Ω p (M ) with compactly overlapping support. The de Rham differential d : Ω p (M ) → Ω p+1 (M ) is a linear differential operator and its formal adjoint is the codifferential δ := d * : Ω p+1 (M ) → Ω p (M ). The d'Alembert operator on p-forms is defined by and it is formally self-adjoint. Because of d 2 = 0 and δ 2 = 0, the d'Alembert operators in different degrees are related by The d'Alembert operators are Green hyperbolic and because of (2.9) the retarded/advanced Green's operators in different degrees are related by (2.10) Finally, we note that the Klein-Gordon-type operators −m 2 : Ω p (M ) → Ω p (M ), where m ∈ R ≥0 is a mass term, are formally self-adjoint Green hyperbolic operators too. ▽

Chain complexes
Chain complexes play a crucial role in formulating and proving our results in this paper. The present subsection contains a brief review of basic aspects of the theory of chain complexes that are necessary for this work. This will in particular allow us to fix the notations and conventions that we employ in the main part of this paper. For more details on chain complexes we refer to [Wei94] and also to [Hov99]. Let us fix a field K of characteristic zero and consider K-vector spaces. In the main sections K will be either the real numbers R or the complex numbers C. A chain complex is a family of vector spaces {V n } n∈Z together with a differential, i.e. a family of linear maps {d n : V n → V n−1 } n∈Z such that d n−1 d n = 0 for all n ∈ Z. To simplify notations, we often denote this data collectively by V and write d : V n → V n−1 for every component of the differential. A chain map f : V → W is a family of linear maps {f n : V n → W n } n∈Z that is compatible with the differentials, i.e. d f n = f n−1 d for all n ∈ Z. We denote by Ch K the category of chain complexes of K-vector spaces with chain maps as morphisms.
The tensor product V ⊗ W ∈ Ch K of two chain complexes V, W ∈ Ch K is defined by for all n ∈ Z, together with the differential obtained by the graded Leibniz rule d(v ⊗ w) := dv ⊗ w + (−1) m v ⊗ dw, for all v ∈ V m and w ∈ W n−m . Note that the ⊗ on the right-hand side of (2.11) is the tensor product of vector spaces. The unit for this tensor product is given by K ∈ Ch K , which we regard as a chain complex concentrated in degree 0 with trivial differential. The tensor product of chain complexes is symmetric via the chain isomorphisms γ : V ⊗ W → W ⊗ V defined by the usual sign-rule γ(v ⊗w) := (−1) m k w ⊗v, for all v ∈ V m and w ∈ W k . Finally, the mapping complex hom(V, W ) ∈ Ch K between two chain complexes V, W ∈ Ch K is defined by for all n ∈ Z, where Lin denotes the vector space of linear maps between vector spaces, together with the "adjoint" differential ∂ : hom(V, W ) n → hom(V, W ) n−1 defined by In summary, this endows Ch K with the structure of a closed symmetric monoidal category.
To every chain complex V ∈ Ch K one can assign its homology H • (V ) = {H n (V )} n∈Z , which is the graded vector space defined by H n (V ) := Ker(d : Quasi-isomorphic chain complexes should be regarded as "being the same", which can be made precise by using techniques from model category theory [Hov99] or ∞-category theory [LurHTT, LurHA]. It is proven in [Hov99] that Ch K carries the structure of a symmetric monoidal model category, whose weak equivalences are the quasi-isomorphisms and fibrations are the degree-wise surjective maps. Because K is by hypothesis a field of characteristic zero, every object in the model category Ch K is both fibrant and cofibrant. Readers who are not familiar with model categories should read the previous statements informally as that "there exists technology to perform a variety of constructions with chain complexes that are compatible with quasi-isomorphisms". In this paper we try to keep the model categorical technicalities to a bare minimum. We refer to [BS19] for a detailed explanation why such techniques are conceptually crucial for formalizing (quantum) gauge theories.
Let us also briefly recall the concept of chain homotopies. A chain homotopy between two chain maps f, g : V → W is a family of linear maps λ = {λ n : V n → W n+1 } n∈Z such that f n − g n = d λ n + λ n−1 d, for all n ∈ Z. This definition can be rephrased very conveniently by using the mapping complexes from (2.12). Note that a chain map f : V → W is precisely a 0-cycle in hom(V, W ) ∈ Ch K , i.e. an element f ∈ hom(V, W ) 0 of degree 0 satisfying ∂f = 0. A chain homotopy between two chain maps f, g : V → W is precisely a 1-chain in hom(V, W ) ∈ Ch K , i.e. an element λ ∈ hom(V, W ) 1 of degree 1, such that ∂λ = f −g. Observe that such chain homotopies exist if and only if the homology class [f −g] ∈ H 0 (hom(V, W )) vanishes. This picture immediately generalizes to higher homotopies: Given two chain homotopies λ, λ ′ ∈ hom(V, W ) 1 between f, g : V → W , then λ − λ ′ is a 1-cycle in hom(V, W ) ∈ Ch K , i.e. ∂(λ − λ ′ ) = 0. A (higher) chain homotopy between λ and λ ′ is a 2-chain χ ∈ hom(V, W ) 2 such that ∂χ = λ − λ ′ . Observe that such (higher) chain homotopies exist if and only if the homology class [λ − λ ′ ] ∈ H 1 (hom(V, W )) vanishes. The pattern for even higher chain homotopies is now evident.
We conclude this subsection by fixing our conventions for shiftings (also called suspensions) of chain complexes. Given any V ∈ Ch K and p ∈ Z, we define where we temporarily used a superscript on d in order to indicate the relevant chain complex. Note that for all p, q ∈ Z, and that V [0] = V . From the definition of the tensor product (2.11), one finds (2.13)

Field and solution complexes
Let M be an oriented and time-oriented globally hyperbolic Lorentzian manifold of dimension m ≥ 2. In this section all chain complexes will be over R, i.e. the relevant category is Ch R . Our aim is to investigate the solution chain complexes for a class of linear gauge field theories on M , which we will obtain from a derived critical locus construction. The following definition will be self-explanatory after Examples 3.2 and 3.3.
concentrated in homological degrees 0 and 1, where is the vector space of sections of a finite-rank real vector bundle F n → M with fiber metric h n , for n = 0, 1, and Example 3.2. Scalar fields on M are described by the field complex concentrated in homological degree 0. The fiber metrics are the ones obtained from the Hodge operator, see Example 2.1. The elements in degree 0 are interpreted as scalar fields Φ ∈ Ω 0 (M ) and triviality of the complex in degree 1 means that there are no gauge transformations, as it should be in a scalar field theory. ▽ Example 3.3. Gauge fields with structure group G = R on M are described by the field complex where d is the de Rham differential. The fiber metrics are the ones obtained from the Hodge operator, see Example 2.1. The elements in degree 0 are interpreted as gauge fields A ∈ Ω 1 (M ) and the elements in degree 1 as gauge transformations ǫ ∈ Ω 0 (M ). The differential d encodes how gauge transformations act on gauge fields, i.e. A → A + dǫ. ▽ Remark 3.4. We would like to mention very briefly that Definition 3.1 admits an obvious generalization to longer complexes is the vector space of sections of a finite-rank real vector bundle F n → M with fiber metric h n and each Q n : F n (M ) → F n−1 (M ) is a linear differential operator. Such generalization is relevant for the description of higher gauge theories, which include gauge transformations between gauge transformations. For example, the complex , and so on. Our results and constructions in this paper apply to this more general case as well, however we decided to focus on 1-gauge theories as in Definition 3.1 in order to improve readability. In particular, our main examples of interest are described by 2-term field complexes, see Examples 3.2 and 3.3. △ In order to encode the dynamics, we consider a formally self-adjoint linear differential operator which we interpret as the equation of motion operator for the fields of the theory. The corresponding quadratic action functional is given by the integration pairing (2.3). This action is gauge-invariant if and only if P satisfies which from now on is always assumed. Because P is formally self-adjoint, it follows that The variation of the action defines a section δ v S : F(M ) → T * F(M ) of the cotangent bundle over F(M ). As in [BS19, Section 3.4], we define the latter as the product complex the smooth dual of the compactly supported field complex F c (M ). Here and in the following we use round brackets to indicate homological degrees. Explicitly, we obtain denotes the inclusion into the first factor and π 2 : F 0 (M ) × F 0 (M ) → F 0 (M ) the projection onto the second factor. The chain map δ v S : F(M ) → T * F(M ) obtained by varying the action then reads explicitly as (3.10) Note the appearance of the equation of motion operator P : F 0 (M ) → F 0 (M ) in the middle vertical arrow. Hence, in order to enforce the equation of motion, we have to intersect δ v S with the zero-section This is the content of the following Definition 3.5. Let F(M ) be a field complex on M and P : F 0 (M ) → F 0 (M ) a formally selfadjoint linear differential operator satisfying (3.8). The corresponding solution complex on M is defined as the derived critical locus of the action functional S in (3.7). Concretely, it is given by the homotopy pullback in the model category Ch R .
Remark 3.6. In this definition we explicitly had to make use of the model category structure on Ch R and the associated concept of homotopy pullbacks, see e.g. [Hov99,Hir03]. We would like to add an informal discussion on the role of homotopy pullbacks for the benefit of those readers who are not familiar with model categories. First, let us note that if (3.12) would be an ordinary categorical pullback, then it would enforce the equation of motion in a strict fashion, i.e. P s 0 = 0. There are however problems with this naive approach, because it is not guaranteed that replacing F(M ) by a quasi-isomorphic chain complex will yield quasi-isomorphic solution complexes Sol(M ). (Recall that quasi-isomorphic chain complexes should be regarded as "being the same".) A homotopy pullback is a suitable deformation (called a derived functor) of the ordinary pullback that does not suffer from this problem. Consequently, our chain complex Sol(M ) from Definition 3.5 is invariant (up to quasi-isomorphisms) under changing F(M ) by quasi-isomorphisms. One should think of our solution complex Sol(M ) as enforcing the equation of motion P s 0 = 0 in only a weak sense, i.e. "up to homotopy". △ Proposition 3.7. A model for the solution complex Sol(M ) from Definition 3.5 is given by (3.13) Proof. The homotopy pullback in (3.12) can be computed by using some basic model category technology, yielding the result in (3.13). The proof for linear Yang-Mills theory in [BS19, Proposition 3.21] generalizes in a straightforward way to our present scenario and hence it will not be repeated.
Example 3.8. For the scalar field complex from Example 3.2, we choose the massive Klein-Gordon operator P = − m 2 : Ω 0 (M ) → Ω 0 (M ). The action in (3.7) is then the usual Klein-Gordon action (3.14) The corresponding solution complex from Proposition 3.7 explicitly reads as The components of this complex admit a physical interpretation in terms of the BRST/BV formalism: • the fields in degree 0 are the scalar fields Φ ∈ Ω 0 (M ); • the fields in degree −1 are the antifields Φ ‡ ∈ Ω 0 (M ).
Note that only the zeroth homology of Sol KG (M ) is non-vanishing. It is given by the ordinary with F = dA ∈ Ω 2 (M ) the field strength. The corresponding solution complex from Proposition 3.7 explicitly reads as (3.17) The components of this complex admit a physical interpretation in terms of the BRST/BV formalism: • the fields in degree 0 are the gauge fields A ∈ Ω 1 (M ); • the fields in degree 1 are the ghost fields c ∈ Ω 0 (M ); • the fields in degrees −1 and −2 are the antifields A ‡ ∈ Ω 1 (M ) and c ‡ ∈ Ω 0 (M ).
The homologies of Sol YM (M ) can be computed explicitly and admit a physical interpretation.
is the first δ-cohomology or equivalently the m−1th de Rham cohomology of M . It captures obstructions to solving the inhomogeneous linear Yang-Mills equation δdA = j with j ∈ Ω 1 δ (M ) a δ-closed 1-form, i.e. δj = 0. For the explicit computation of H −1 (Sol YM (M )) one uses standard techniques from the theory of normally hyperbolic operators [BGP07,Bar15] in order to prove that δdA = j admits a solution A if and only if j = δζ is δ-exact.
is the zeroth δ-cohomology or equivalently the m-th de Rham cohomology of M . This is trivial because every globally hyperbolic Lorentzian manifold is diffeomorphic to a product manifold M ∼ = R × Σ.
We in particular observe that Sol YM (M ) can not be quasi-isomorphic to a chain complex concentrated in degree 0, hence it contains more refined information than the vector space of gauge equivalence classes of linear Yang-Mills solutions, i.e. the zeroth homology H 0 (Sol YM (M )). It is the latter that is traditionally considered in the AQFT literature, see e.g. [ In these examples there exist two distinct types of chain homotopies (called retarded and advanced) that trivialize the shifted Poisson structure, which play an analogous role to the retarded and advanced Green's operators in ordinary field theory, see e.g. [BGP07,BDH13]. Taking the difference between a compatible pair of retarded and advanced trivializations allows us to define an unshifted Poisson structure on Sol(M ), which is the necessary ingredient for canonical commutation relations (CCR) quantization in Section 5.
Both the shifted and unshifted Poisson structures will be defined on the smooth dual of the solution complex from Proposition 3.7, which should be interpreted as a chain complex of linear observables.
where the subscript c denotes compactly supported sections. The integration pairings (2.3) define evaluation chain maps where j pc/fc is the evident extension of the chain map (4.4) to sections with past/future compact support.
The following are some simple properties of retarded/advanced trivializations.
Remark 4.10. We would like to emphasize that our Definition 4.7 of unshifted Poisson structures leaves one important question unanswered: Do compatible pairs of retarded/advanced trivializations exist? We do already know from Lemma 4.5 b) that, provided they exist, retarded/advanced trivializations are unique up to homotopy, and so are their associated unshifted Poisson structures, see Corollary 4.9. Note that such questions are analogs of existence and uniqueness for Green's operators in ordinary field theory. We shall now investigate these issues in detail for Klein-Gordon and linear Yang-Mills theory. This will in particular clarify the relationship between retarded/advanced trivializations and retarded/advanced Green's operators. Proof. Recall from Section 2.1 that the (extended) retarded/advanced Green's operator G ± : Ω 0 pc/fc (M ) → Ω 0 pc/fc (M ) for − m 2 satisfies (4.17) and hence setting Λ ± 0 = G ± defines a retarded/advanced trivialization Λ ± ∈ hom(L KG pc/fc (M ), L KG pc/fc (M )) 1 . Uniqueness follows from Lemma 4.5 b) and the fact that hom(L KG pc/fc (M ), L KG pc/fc (M )) 2 = 0 is the zero vector space.
Because of (2.6), the unique retarded and advanced trivializations obtained above form a compatible pair in the sense of Definition 4.6. The corresponding unshifted Poisson structure from Definition 4.7 then reads as

Linear Yang-Mills theory
define a compatible pair of retarded/advanced trivializations for linear Yang-Mills theory.
Proof. This follows immediately from the properties of Green's operators stated in Section 2.1, see in particular Example 2.1.
Remark 4.13. In contrast to the example of Klein-Gordon theory from Section 4.1, the retarded/advanced trivializations of linear Yang-Mills theory are not unique, but only unique up to homotopy, see Lemma 4.5. Any other retarded/advanced trivialization Λ ± differs from our Λ ± above by the differential of a 2-chain λ ± ∈ hom(L YM pc/fc (M ), L YM pc/fc (M )) 2 . Explicitly, the three non-zero components of Λ ± read as : Ω 0 pc/fc (M ) −→ Ω 1 pc/fc (M ) , The unshifted Poisson structure (see Definition 4.7) that corresponds to our compatible pair of retarded/advanced trivializations Λ ± from Proposition 4.12 reads as To conclude this section, we would like to emphasize that any other choice of a compatible pair of retarded/advanced trivializations Λ ± (see Remark 4.13 for a concrete description) defines an unshifted Poisson structure τ = τ YM + ∂ρ that agrees with (4.26) up to homotopy, see Corollary 4.9. We shall prove in Proposition 5.3 that the quantization of two homotopic Poisson structures yields quasi-isomorphic observable algebras, i.e. the quasi-isomorphism type of the resulting quantum theory depends only on the uniquely defined homology classes [Λ ± ] ∈ H 1 (hom(L YM pc/fc (M ), L YM pc/fc (M ))).

Quantization
The goal of this section is to develop a chain complex analog of the usual canonical commutation relations (CCR) quantization of vector spaces endowed with Poisson structures, see e.g. [BDH13]. The input of our construction is a pair (V, τ ) consisting of a chain complex V ∈ Ch R and a chain map τ : V ∧ V → R. We shall call (V, τ ) an unshifted Poisson complex. The output of our construction is a differential graded unital and associative * -algebra CCR(V, τ ) over the field of complex numbers C that implements the canonical commutation relations determined by τ . We shall investigate homotopical properties of this quantization prescription and in particular prove that, up to quasi-isomorphism, the quantization CCR(V, τ ) does only depend on the quasiisomorphism type of (V, τ ) and on the homology class [τ ] ∈ H 0 (hom( 2 V, R)) of τ . In the context of our examples from Section 4, this means that both Klein-Gordon theory and linear Yang-Mills theory can be consistently quantized by our methods.
Let us now explain in some detail the CCR quantization CCR(V, τ ) of an unshifted Poisson complex (V, τ ). We denote by T ⊗ C V the free differential graded unital and associative * -algebra generated by V ∈ Ch R . Concretely, T ⊗ C V is given by The CCR quantization is defined as the quotient by the two-sided differential graded * -ideal I (V,τ ) ⊆ T ⊗ C V generated by the (graded) canonical commutation relations for all homogeneous elements v 1 , v 2 ∈ V with degrees denoted by |v 1 |, |v 2 | ∈ Z. We note that CCR quantization is functorial for the following natural choices of categories: • PoCh R denotes the category of unshifted Poisson complexes, i.e. objects are pairs (V, τ ) consisting of a chain complex V ∈ Ch R and a chain map τ : V ∧ V → R and morphisms f : • dg * Alg C denotes the usual category of differential graded unital and associative * -algebras.
Remark 5.1. Every ordinary Poisson vector space (V, τ ) defines an unshifted Poisson complex whose underlying chain complex is concentrated in degree 0. In such cases our CCR quantization CCR(V, τ ) yields a differential graded * -algebra concentrated in degree 0, which coincides with the usual CCR algebra from the non-homotopical framework, see e.g. [BDH13]. △ For our homotopical analysis of CCR quantization, we endow both PoCh R and dg * Alg C with the structure of a homotopical category in the sense of [DHKS04,Rie14]. This is a more flexible framework than model category theory, which is very convenient for our purposes because PoCh R is not a model category as it is not cocomplete. Similarly to model category theory, a homotopical category is a category with a choice of weak equivalences (containing all isomorphisms and satisfying the so-called 2-of-6 property), however there is no need to introduce compatible classes of fibrations and cofibrations or to require the category to be bicomplete. In our context, we introduce the following canonical homotopical category structures on PoCh R and dg * Alg C .
(ii) A morphism κ : A → A ′ in dg * Alg C is a weak equivalence if its underlying chain map is a quasi-isomorphism in Ch C .
The next result shows that the CCR functor (5.4) has very pleasant homotopical properties, which in particular ensure that our examples of linear gauge theories from Section 4 can be quantized consistently. The proof of the following proposition is slightly technical and hence it will be carried out in detail in Appendix A.
Then there exists a zig-zag of weak equivalences in dg * Alg C .
In our context of linear gauge theories from Section 4, we immediately obtain the following crucial result as a direct consequence of Proposition 5.3 and Corollary 4.9. As observed in Examples 5.5 and 5.6, our homotopy theoretical constructions naturally define differential graded * -algebras of quantum observables for each spacetime M . As a consequence, the relevant variants of AQFT to describe such models should take values in the model category Ch C of chain complexes in contrast to the usual category Vec C of vector spaces. Based on the recent operadic approach to AQFT [BSW17,BSW19a], such algebraic structures were systematically investigated in [BSW19b]. One of the outcomes of these studies are various models for homotopy AQFTs, which are obtained by different resolutions of the relevant operad. Since in the present paper our ground field C has characteristic 0, the strictification theorem of [BSW19b] implies that every homotopy AQFT can be strictified and hence all possible variants of homotopy AQFT are equivalent. For our concrete examples investigated in the present paper, the following semi-strict model for homotopy AQFTs is most convenient.
Definition 6.1. A homotopy AQFT on Loc is a functor A : Loc → dg * Alg C such that the following axioms hold true: In these examples f * is simply given by pullback of differential forms. △ Remark 6.4. Using as in Remark 6.2 the forgetful functor U M : Loc/M → Loc, all functors and natural transformations on Loc that we introduced above can be restricted to the slice category Loc/M , for each M ∈ Loc. This restricted data is sufficient when one attempts to construct only a homotopy AQFT on a fixed M ∈ Loc, in contrast to a homotopy AQFT on Loc. △ Our approach to construct unshifted Poisson structures (Definition 4.7) in terms of (compatible pairs of) retarded/advanced trivializations (Definition 4.4) has to be supplemented by a suitable naturality axiom. Because the strength of our final result will depend on whether we work with Loc or a slice category Loc/M , for some M ∈ Loc, we shall state our definitions and results below for both cases. where L : C → Ch R is the functor assigning chain complexes of linear observables. A C-natural homotopy between two C-natural unshifted Poisson structures τ, τ ∈ hom( 2 L, R) 0 is a 1-chain ρ ∈ hom( 2 L, R) 1 , such that τ − τ = ∂ρ.
Remark 6.9. We decided to state Definition 6.8 in a rather abstract form because this will become useful later. From a more concrete perspective, the data of a C-natural unshifted Poisson structure τ ∈ hom( 2 L, R) 0 is given by a family Proof. Item a) is immediate because the definition of the unshifted Poisson structure in (4.10) involves only natural maps. Item c) follows from item b) and Corollary 4.9. It thus remains to prove item b), which follows immediately from the fact that the slice category Loc/M has a terminal object (id : M → M ), hence (Loc/M ) op has an initial object. The limit in (6.10) is then isomorphic to the chain complex hom( 2 L(M ), R) corresponding to this object.
Remark 6.11. It is currently unclear to us if the analog of Lemma 6.10 c) also holds true for the category Loc. Let us explain this issue in more detail. Suppose that Λ ± , Λ ± are two Loc-natural compatible pairs of retarded/advanced trivializations and denote the corresponding Loc-natural unshifted Poisson structures by τ, τ . By Corollary 4.9, we obtain that for every M ∈ Loc there exists a 1-chain ρ M ∈ hom( 2 L(M ), R) 1 such that τ M − τ M = ∂ρ M . However, it is unclear whether such homotopies can be chosen to be Loc-natural as Loc has no terminal object. (The terminal object in Loc/M was crucial to prove Lemma 6.10 b) and hence c).) As a consequence, it is currently unclear to us if the particular model for linear quantum Yang-Mills theory that we will construct below is, up to natural weak equivalences, the only possibility within our approach. In particular, we can not exclude the existence of a Loc-natural compatible pair of retarded/advanced trivializations different from the one in Proposition 6.7 b), that leads to a non-homotopic Loc-natural unshifted Poisson structure and hence potentially to a non-equivalent quantization. △ Example 6.12. Let us apply our general results to Klein-Gordon theory, see Examples 3.2 and 3.8 as well as Section 4.1. The Loc-natural compatible pair of retarded/advanced trivializations from Proposition 6.7 a) defines via Lemma 6.10 a) a Loc-natural unshifted Poisson structure τ KG , whose components τ KG M , for M ∈ Loc, are given concretely by (4.18). Due to the component-wise uniqueness result for retarded/advanced trivializations for Klein-Gordon theory in Proposition 4.11, it follows that τ KG is unique too. Hence, in the case of Klein-Gordon theory we obtain stronger results than in the general Lemma 6.10. ▽ Example 6.13. Let us now apply our general results to linear Yang-Mills theory, see Examples 3.3 and 3.9 as well as Section 4.2. The Loc-natural compatible pair of retarded/advanced trivializations from Proposition 6.7 b) defines via Lemma 6.10 a) a Loc-natural unshifted Poisson structure τ YM , whose components τ YM M , for M ∈ Loc, are given concretely by (4.26). Unfortunately, as explained in Remark 6.11, we are currently unable to exclude the existence of other Loc-natural choices of compatible pairs of retarded/advanced trivializations that define nonhomotopic Loc-natural unshifted Poisson structures. The situation gets much better when we work on a slice category Loc/M , for any M ∈ Loc. In this case Proposition 6.7 b) restricts to a Loc/M -natural compatible pair of retarded/advanced trivializations and Lemma 6.10 a) defines a Loc/M -natural unshifted Poisson structure τ YM . By Lemma 6.10 c), we know that any other to the category of differential graded * -algebras.
In order to analyze homotopical properties of this construction, we endow both functor categories Fun(C, PoCh R ) and Fun(C, dg * Alg C ) with the structure of a homotopical category [DHKS04,Rie14] in which weak equivalences are so-called natural weak equivalences.
Definition 6.14. Let C be either Loc or Loc/M , for any M ∈ Loc.
(i) A morphism in Fun(C, PoCh R ) (i.e. a natural transformation) is a natural weak equivalence if all its components are weak equivalences in PoCh R , see Definition 5.2.
(ii) A morphism in Fun(C, dg * Alg C ) (i.e. a natural transformation) is a natural weak equivalence if all its components are weak equivalences in dg * Alg C , see Definition 5.2.
(iii) Let hAQFT(C) ⊆ Fun(C, dg * Alg C ) denote the full subcategory of functors satisfying the homotopy AQFT axioms from Definition 6.1. A morphism in hAQFT(C) is a weak equivalence if and only if it is a natural weak equivalence in Fun(C, dg * Alg C ).
Proof. Item a) is an immediate consequence of the component-wise definition of natural weak equivalences in Definition 6.14 and the result in Proposition 5.3 a) that the CCR functor is a homotopical functor.
Let us now focus on item b). By Proposition 5.3 b) and the explicit construction in Proposition A.3, we obtain for each object M ∈ C a zig-zag Together with Lemma 6.10 c), Proposition 6.16 b) implies the following important result.
Corollary 6.17. Fix any M ∈ Loc and suppose that Λ ± and Λ ± are two Loc/M -natural compatible pairs of retarded/advanced trivializations. Denote the corresponding Loc/M -natural unshifted Poisson structures from Lemma 6.10 a) by τ and τ . Then the two functors A := CCR(L, τ ) and A := CCR(L, τ ) are equivalent via a zig-zag of natural weak equivalences in Fun(Loc/M , dg * Alg C ).
The next lemma provides conditions on (L, τ ) : C → PoCh R which imply that A := CCR(L, τ ) : C → dg * Alg C fulfills the homotopy AQFT axioms from Definition 6.1. Proof. Item a) is a direct consequence of the canonical commutation relations in (5.3). Item b) follows from the fact that CCR is a homotopical functor, see Proposition 5.3 a).
We are now ready to state and prove the main result of the present paper.
Theorem 6.19. a) Let τ KG denote the unshifted Poisson structure defined by Lemma 6.10 a) from the unique Loc-natural compatible pair of retarded/advanced trivializations for Klein-Gordon theory, see Proposition 6.7 a). Then the functor A KG := CCR(L KG , τ KG ) : Loc → dg * Alg C is a homotopy AQFT on Loc, i.e. A KG ∈ hAQFT(Loc). This homotopy AQFT on Loc is determined uniquely up to natural weak equivalences by our construction. b) Let τ YM denote the unshifted Poisson structure defined by Lemma 6.10 a) from the Locnatural compatible pair of retarded/advanced trivializations for linear Yang-Mills theory, see Proposition 6.7 b). Then the functor A YM := CCR(L YM , τ YM ) : Loc → dg * Alg C is a homotopy AQFT on Loc, i.e. A YM ∈ hAQFT(Loc). The restriction A YM M := A YM U M ∈ hAQFT(Loc/M ) given in Remark 6.2 defines a homotopy AQFT on each M ∈ Loc. These homotopy AQFTs on M are determined uniquely up to natural weak equivalences by our construction.
Proof. Item a): Example 6.12 defines a functor (L KG , τ KG ) : Loc → PoCh R and hence by postcomposition with CCR a functor A KG := CCR(L KG , τ KG ) : Loc → dg * Alg C . It remains to prove that this functor satisfies the homotopy AQFT axioms from Definition 6.1, which we shall do by checking the sufficient conditions on (L KG , τ KG ) from Lemma 6.18. We deduce from the explicit expressions for τ KG in (4.18) and the support properties of retarded/advanced Green's operators (see Section 2.1) that the hypothesis of Lemma 6.18 a) is fulfilled, hence A KG satisfies Einstein causality.
In order to prove time-slice, recall from Example 5. In contrast to the situation for Klein-Gordon theory explained in Remark 6.20, our model in Theorem 6.19 b) for linear Yang-Mills theory as a homotopy AQFT on Loc is not naturally weakly equivalent to existing models in the literature that consider only gauge-invariant onshell observables, see e.g. [SDH14,BDS14,BDHS14,FL16,Ben16,BSS16]. This is because, on a generic M ∈ Loc, the complex of linear observables L YM (M ) has non-trivial homology in degrees n = −1, 0, 1, while the usual models in the literature consider only its zeroth homology. In the terminology of the BRST/BV formalism, one can say that our description of linear Yang-Mills theory as a homotopy AQFT A YM ∈ hAQFT(Loc) takes fully into account all ghost and antifield observables, while the traditional models consider only the 0-truncation of the antifield number 0 sector of the theory. In particular, notice that the difference between A YM (M ) and CCR(H 0 (L YM (M )), τ YM M ), for a generic M ∈ Loc, is already visible on the level of the zeroth homology: The * -algebra CCR(H 0 (L YM (M )), τ YM M ) is generated only by linear gauge-invariant on-shell observables, while the * -algebra H 0 (A YM (M )) contains also classes that are obtained by multiplying in A YM (M ) an equal number of ghost field and antifield linear observables. △ Proposition A.2. Heis : PoCh R → dg * uLie C is a homotopical functor. Together with Lemma A.1, this proves Proposition 5.3 a).
Proof. We construct an explicit object H (V,τ,ρ) ∈ dg * uLie C that allows us to exhibit the desired zig-zag of weak equivalences. Let us introduce the acyclic chain complex together with the unit ½ := 0 ⊕ 0 ⊕ 1 and the Lie bracket For any real number s ∈ R, we let I s ⊆ H (V,τ,ρ) be the differential graded unital Lie * -algebra ideal generated by the two relations 0 ⊕ x ⊕ 0 = 0 ⊕ 0 ⊕ s , 0 ⊕ y ⊕ 0 = 0 . is isomorphic to the Heisenberg Lie algebra of (V, τ + s ∂ρ) ∈ PoCh R . We still have to show that the quotient map π s : H (V,τ,ρ) −→ Heis(V, τ + s ∂ρ) (A.9) is a weak equivalence in dg * uLie C . From the explicit form of the relations in (A.7), we observe that π s = id V ⊕ q s with q s : D ⊕ C → C given by q s : (c 1 x + c 2 y) ⊕ c 3 → s c 1 + c 3 . This is clearly a quasi-isomorphism in Ch C , hence (A.9) is a weak equivalence for any s ∈ R. The desired zig-zag follows by taking s = 0 and s = 1.