Quantization of Lorentzian free BV theories: factorization algebra vs algebraic quantum field theory

We construct and compare two alternative quantizations, as a time-orderable prefactorization algebra and as an algebraic quantum field theory valued in cochain complexes, of a natural collection of free BV theories on the category of $m$-dimensional globally hyperbolic Lorentzian manifolds. Our comparison is realized as an explicit isomorphism of time-orderable prefactorization algebras. The key ingredients of our approach are the retarded and advanced Green's homotopies associated with free BV theories, which generalize retarded and advanced Green's operators to cochain complexes of linear differential operators.


Introduction and summary
Several mathematical axiomatizations of quantum field theory (QFT) on Lorentzian manifolds have been proposed in the literature, such as algebraic quantum field theories (AQFTs) [BFV03,FV15,BSW21] and time-orderable prefactorization algebras [BPS20], i.e. a Lorentzian variant of prefactorization algebras [CG17,CG21].These two approaches are a priori quite different.For instance, while the former emphasizes the algebraic structure carried by the quantum observables on each spacetime, the latter focuses on their time-ordered products.The differences become even more striking when one tries to construct simple QFT models, such as the free Klein-Gordon field of mass m ≥ 0: while the corresponding time-orderable prefactorization algebra is constructed out of the (−1)-shifted Poisson structure (antibracket) τ (−1) (ϕ ⊗ ϕ ‡ ) := M ϕϕ ‡ vol M only, the corresponding AQFT relies crucially also on the retarded and advanced Green's operators G ± for the Klein-Gordon operator + m 2 through the unshifted Poisson structure τ (0) (ϕ 1 ⊗ ϕ 2 ) := Because of these differences it is interesting to compare time-orderable prefactorization algebras and AQFTs.This task was undertaken first in a model-based approach by [GR20] and then in a model-independent fashion by [BPS20].In [GR20] it is shown that the time-orderable prefactorization algebra and the AQFT of the free Klein-Gordon field encode equivalent information as a consequence of the time-slice axiom, i.e. the property that any spacetime embedding whose image contains a Cauchy surface of the codomain induces an isomorphism at the level of quantum observables.(The results of [GR20] can be adapted with minor modifications to encompass any field theoretic model that is ruled by a Green hyperbolic operator.)In [BPS20] a model-independent comparison is developed in the form of an equivalence (actually isomorphism) between the categories of time-orderable prefactorization algebras and of AQFTs, both satisfying the time-slice axiom (and an additional technical requirement, called additivity, that is fulfilled by many examples).
Unfortunately, the results of [GR20] and [BPS20] do not cover the examples of linear gauge theories.On the one hand, the equation of motion of a linear gauge theory (with gauge transformations acting non-trivially) must be degenerate.In particular, the corresponding linear differential operator is not a Green hyperbolic operator, see [Bar15] and also Definition 2.8.As a consequence, the results of [GR20] cannot be applied directly.On the other hand, linear gauge theories are most naturally encoded by cochain complexes in the spirit of the BV formalism, see [CG17,CG21,Hol08,FR12,FR13,BS19].In this context a weaker version of the timeslice axiom holds, where isomorphisms are replaced by quasi-isomorphisms, see [BBS20] and also Examples 4.4 and 4.7.Motivated by this fact, linear gauge theories on Lorentzian manifolds are realized by means of time-orderable prefactorization algebras or AQFTs that take values in the ∞-category Ch C of cochain complexes with equivalences given by quasi-isomorphisms, see Definitions 2.11 and 2.9 and also Remark 2.10.While we are currently not able to upgrade the model-independent comparison of [BPS20] to the case where the target is the ∞-category Ch C , with the present paper we extend the results of [GR20] to linear gauge (and also higher gauge) theories.The key ingredient to achieve this goal is a generalization of Green hyperbolic operators, namely the recently developed Green hyperbolic complexes [BMS22].In contrast to Green hyperbolic operators, Green hyperbolic complexes cover many important examples of linear gauge theories, see [BMS22] and also Examples 3.6, 3.7 and 3.8.Their key feature is that they admit retarded and advanced Green's homotopies Λ ± , generalizing the familiar retarded and advanced Green's operators G ± for Green hyperbolic operators.
The input of our construction is a free BV theory (F, Q, (−, −), W ) on an m-dimensional oriented and time-oriented globally hyperbolic Lorentzian manifold M ∈ Loc m , consisting of a complex of linear differential operators (F, Q) with a compatible (−1)-shifted fiber metric (−, −) and a (formally self-adjoint) Green's witness, see Definitions 3.1, 3.3 and 3.5.Let us provide some interpretation of these data and some information about the structures that can be defined out of it.In the spirit of the BV formalism one may think of the graded vector bundle F as encoding both gauge and ghost fields, as well as the respective antifields.In the same spirit the differential Q, which is degree-wise a linear differential operator, encodes both the action of gauge transformations and the equation of motion.The compatible (−1)-shifted fiber metric (−, −) is a suitable generalization of the more familiar concept of a fiber metric on a vector bundle.(−, −) is closely related to the antibracket from the BV formalism in the sense that, upon integration, it defines the (−1)-shifted Poisson structure τ (−1) on the 1-shift F c (M )[1] ∈ Ch R of the cochain complex of compactly supported smooth sections of (F, Q), see (3.13).Finally, the role of the Green's witness W is to give rise to the Green hyperbolic operator P := Q W + W Q, which allows one to find particularly simple retarded and advanced Green's homotopies Λ ± := W G ± , where G ± denote the retarded and advanced Green's operator for P .In this sense W "witnesses" the fact that (F, Q) is a Green hyperbolic complex.Taking the difference of Λ + and Λ − defines the retarded-minus-advanced cochain map Λ := Λ + − Λ − and taking their average defines the Dirac homotopy Λ D := 1 2 (Λ + + Λ − ), which generalize the familiar retarded-minus-advanced G := G + − G − and Dirac G D := 1 2 (G + + G − ) propagators.In combination with the (−1)-shifted fiber metric (−, −), Λ and Λ D define, upon integration, the unshifted Poisson structure τ (0) and respectively the Dirac pairing τ D on the cochain complex F c (M )[1] ∈ Ch R of compactly supported smooth sections, see (3.15) and (3.17).The (−1)-shifted Poisson structure τ (−1) plays a crucial role in the first step of our construction (quantization as a time-orderable prefactorization algebra), the unshifted Poisson structure τ (0) in the second step (quantization as an AQFT) and the Dirac pairing τ D in the last step (comparison).
In the first step, which is carried out in Subsection 4.1, we construct a time-orderable prefactorization algebra F ∈ tPFA m out of a collection (F M , Q M , (−, −) M , W M ) M ∈Locm of free BV theories that is natural with respect to the morphisms f : M → N in Loc m (see Appendix A for the technical details).The first part of this construction relies only on the complexes of linear differential operators (F M , Q M ) and on the compatible (−1)-shifted fiber metrics (−, −) M , for all M ∈ Loc m .These data are used to define the (−1)-shifted Poisson structures τ (−1) , whose BV quantization provides the time-orderable prefactorization algebra F of interest to us.Explicitly, from τ (−1) we define the BV Laplacian ∆ BV on the symmetric algebra Sym(F c (M )[1]) ∈ dgCAlg C , see (4.3), and then we deform the original differential Q to the quantized differential Q := Q + i ∆ BV .Even though Q is not compatible with the commutative multiplication µ of the symmetric algebra Sym(F c (M )[1]) ∈ dgCAlg C , it is compatible with the time-ordered products constructed out of µ, see Proposition 4.2.Hence, by defining for all M ∈ Loc m the cochain complexes F(M ) := (Sym(F c (M )[1]), Q ) ∈ Ch C that consist of the graded vector space underlying Sym(F c (M )[1]) with the quantized differential Q , we obtain the time-orderable prefactorization algebra F with time-ordered products constructed out of the symmetric algebra multiplication µ.At this point it is unclear whether F fulfills the timeslice axiom.The Green's witnesses W M become crucial for this purpose, see Theorem 3.13 and Proposition 4.3.
In the second step, which is carried out in Subsection 4.2, we construct an AQFT A ∈ AQFT m out of the same data.Explicitly, instead of using τ (−1) to deform the differential, here one uses the unshifted Poisson structure τ (0) to deform the commutative multiplication µ of the symmetric algebra Sym(F c (M )[1]) ∈ dgCAlg C to the (in general non-commutative) Moyal-Weyl star product µ , see (4.13).The deformed multiplication µ is compatible with the original differential Q and with the pushforward of compactly supported sections along Loc m -morphisms.Hence, we obtain the AQFT A by defining for all M ∈ Loc m the differential graded algebras consist of (the cochain complex underlying) Sym(F c (M )[1]) with the Moyal-Weyl star product µ and the unit ½ ∈ Sym(F c (M )[1]), and extending the pushforward of compactly supported sections.
In the last step, which is carried out in Subsection 4.3, we compare the time-orderable prefactorization algebra F ∈ tPFA m and the AQFT A ∈ AQFT m obtained in the previous steps.Explicitly, we construct a comparison isomorphism T : F → F A in tPFA m between F ∈ tPFA m and the time-orderable prefactorization algebra F A ∈ tPFA m associated with A ∈ AQFT m , whose time-ordered products are constructed out of the Moyal-Weyl star product µ .(F A is just the evaluation on A of the functor AQFT m → tPFA m from [BPS20].)The comparison isomorphism T := exp( i ∆ D ) is defined as the exponential of the Dirac Laplacian ∆ D , which is obtained from the Dirac pairing τ D , see Theorem 4.9.In particular, we show that T intertwines the quantized differential Q with the original symmetric algebra differential Q and the timeordered products constructed out of the original symmetric algebra multiplication µ with those constructed out of the quantized multiplication µ .
The outline of the rest of the paper is the following.Section 2 contains the background material needed later on.In particular, Subsection 2.1 reviews some basic aspects of the theory of cochain complexes Ch K over a field K of characteristic zero; Subsection 2.2 describes the extension of (anti-)symmetric pairings τ of degree p ∈ Z on a cochain complex V ∈ Ch K to suitable bi-derivations {{−, −}} τ and, in the symmetric case, to suitable Laplacians ∆ τ on the symmetric algebra Sym V ∈ dgCAlg K ; Subsection 2.3 recalls some relevant concepts from Lorentzian geometry and Green hyperbolic operators; Subsection 2.4 reviews the concepts of time-orderable prefactorization algebras and AQFTs valued in cochain complexes Ch C , including the Einstein causality and time-slice axioms (the latter in the form of a quasi-isomorphism).Section 3 focuses on the concept of a Green's witness and on the structures that can be constructed out of it.More in detail, Subsection 3.1 recalls the concepts of a complex of linear differential operators (F, Q), of a compatible (−1)-shifted fiber metric (−, −) and of a (formally self-adjoint) Green's witness W , which together form a free BV theory (F, Q, (−, −), W ) on M ∈ Loc m , and out of these data it constructs the (−1)-shifted Poisson structure τ (−1) , the unshifted Poisson structure τ (0) and the Dirac pairing τ D ; Subsection 3.2 investigates the properties of the structures τ (−1) , τ (0) and τ D associated with a natural collection (F M , Q M , (−, −) M , W M ) M ∈Locm of free BV theories, proving in particular classical analogs of the Einstein causality and time-slice axioms, see Theorem 3.13.The core of the paper is Section 4, which is devoted to the construction and comparison of two alternative quantizations of a natural collection (F M , Q M , (−, −) M , W M ) M ∈Locm of free BV theories.The starting point of both quantization schemes is the symmetric algebra Sym(F c (M )[1]) ∈ dgCAlg C , where F c (M )[1] ∈ Ch R denotes the 1-shift of the cochain complex of compactly supported smooth sections of the complex of linear differential operators T intertwines the quantized differential Q and original (i.e.constructed out of µ) time-ordered products of F ∈ tPFA m with the original differential Q and quantized (i.e.constructed out of µ ) time-ordered products of the time-orderable prefactorization algebra F A ∈ tPFA m associated with A ∈ AQFT m according to [BPS20].Appendix A discusses some technical details about naturality of vector bundles, fiber metrics and differential operators which we require to introduce the concept of natural free BV theories.

Cochain complexes
We review some basic aspects of the theory of cochain complexes to fix our notation and conventions.More details on the well-known topics recalled here are covered by the classical literature, see e.g.[Wei94,Hov99].Let us fix a field K of characteristic zero.In the main part of this paper K will be either the field R of real numbers or the field C of complex numbers.Definition 2.1.
, for all n ∈ Z.We denote by Ch K the category whose objects are cochain complexes and whose morphisms are cochain maps.

The tensor product
for all n ∈ Z, and of the differential Q ⊗ given by the graded Leibniz rule for all homogeneous v ∈ V and w ∈ W , where |−| denotes the degree.The monoidal unit of the tensor product is given by K ∈ Ch K , regarded as a cochain complex concentrated in degree zero with trivial differential.The symmetric braiding is given by the cochain maps γ : V ⊗W → W ⊗V in Ch K that are defined by the Koszul sign rule for all homogeneous v ∈ V and w ∈ W .The internal hom [V, W ] ∈ Ch K is the cochain complex that consists of for all n ∈ Z, where Hom K denotes the vector space of linear maps, and of the differential ∂ defined by To each cochain complex V ∈ Ch K one can assign its cohomology H • (V ) = (H n (V )) n∈Z , that is the graded vector space defined degree-wise by , for all n ∈ Z.The compatibility of cochain maps with differentials entails that cohomology extends to a functor H • from Ch K to the category of graded vector spaces.A cochain map f : In many circumstances quasi-isomorphic cochain complexes should be regarded as "being the same", which can be made precise by using techniques from model category theory.It is proven in [Hov99] that Ch K carries the structure of a closed symmetric monoidal model category, whose weak equivalences are the quasi-isomorphisms and whose fibrations are the degree-wise surjective cochain maps.
Remark 2.2.Let us briefly recall how one may interpret the cohomology of the internal hom [V, W ] ∈ Ch K between cochain complexes V, W ∈ Ch K in terms of higher cochain homotopies.Given two n-cocycles f, g ∈ Ker(∂ n ) in [V, W ], one defines a cochain homotopy from f to g as an ) coincide if and only if a cochain homotopy from f to g exists.In particular, for n = 0 one recovers the ordinary concept of cochain homotopies between two cochain maps f, g : V → W in Ch K .△ Let us also fix our convention for shifts of cochain complexes.For a cochain complex V ∈ Ch K and an integer q ∈ Z, we define the q-shift V [q] ∈ Ch K of V as the cochain complex consisting of V [q] n := V q+n , for all n ∈ Z, and of the differential for all p, q ∈ Z, and that V [0] = V .Furthermore, recalling the definition of the tensor product (2.1), one obtains natural cochain isomorphisms V [q] ∼ = K[q] ⊗ V for all q ∈ Z.

Extension of pairings to symmetric algebras
In this paper we will encounter various types of pairings τ ∈ [V ⊗ V, K] p of degree p ∈ Z on a cochain complex V ∈ Ch K .These pairings are either symmetric or anti-symmetric, i.e.
with s = +1 in the symmetric case and s = −1 in the anti-symmetric case, where γ denotes the symmetric braiding of Ch K .In particular, we shall consider shifted and also unshifted (i.e.0-shifted) (linear) Poisson structures as defined below.
Definition 2.3.A p-shifted (linear) Poisson structure on a cochain complex V ∈ Ch K consists of a symmetric (respectively, anti-symmetric) pairing τ ∈ [V ⊗ V, K] p of odd (respectively, even) degree p ∈ Z that is closed ∂τ = 0 with respect to the internal hom differential (2.3).
The aim of this subsection is to describe an extension of such pairings to suitable bi-derivations and, in the symmetric case, to suitable Laplacians on the symmetric algebra Sym V ∈ dgCAlg K .The latter is the commutative differential graded algebra defined by Sym V = ∞ n=0 Sym n V , with unit element ½ := 1 ∈ Sym 0 V = K and multiplication for all n, n ′ ≥ 0 and all v 1 , . . ., v n , v ′ 1 , . . ., v ′ n ′ ∈ V .(By convention, the length n = 0 corresponds to the unit ½.) Definition 2.4.Given an (anti-)symmetric pairing τ ∈ [V ⊗ V, K] p of degree p, we define as the unique graded linear map of degree p that fulfills the following conditions: with s = +1 in the symmetric case and s = −1 in the anti-symmetric case; (iii) for all homogeneous a ∈ Sym V , {{a, −}} τ : Sym V → Sym V ⊗ Sym V is a graded derivation of degree |a| + p with respect to the (Sym V )-module structure on Sym V ⊗ Sym V given by multiplication on the second tensor factor, i.e.
(2.9) Furthermore, given two cochain complexes V, W ∈ Ch K endowed with (anti-)symmetric pairings τ ∈ [V ⊗V, K] p and ω ∈ [W ⊗W, K] p of degree p and a cochain map (2.10) Remark 2.5.A p-shifted (linear) Poisson structure τ on V can be extended to a p-shifted Poisson bracket {−, −} τ on Sym V .Indeed, from Definitions 2.3 and 2.4 it follows that defines a p-shifted Poisson bracket, i.e. a graded linear map of degree p that is closed ∂{−, −} τ = 0, symmetric (respectively, anti-symmetric) for p odd (respectively, even) and fulfills the graded Leibniz rule and the Jacobi identity.△ Definition 2.6.Given a symmetric pairing τ ∈ [V ⊗ V, K] p of degree p, we define the Laplacian as the unique graded linear map of degree p that fulfills the following conditions: (2.13) The defining properties of ∆ τ imply the explicit formula for all n ≥ 1 and all homogeneous v 1 , . . ., v n ∈ V , where • means to omit the corresponding factor.Furthermore, for p even, iterating (2.13) and observing that both ∆ τ ⊗ id and id ⊗ ∆ τ graded commute with {{−, −}} τ , one finds that, for all n ≥ 1, where ∆ τ ⊗ := ∆ τ ⊗ id + id ⊗ ∆ τ .(For n = 1 this recovers (2.13).For p odd, the left-hand side vanishes identically for n ≥ 2, see (2.17) below.)Taking also (2.9) into account, one shows that degrees p and p ′ respectively, the explicit formula (2.14) for the Laplacian entails

Lorentzian geometry and Green's operators
In this subsection we recall some relevant concepts from Lorentzian geometry and Green hyperbolic differential operators.We refer to [Bar15,BGP07,ONe83] for an in-depth introduction to these topics.
A Lorentzian manifold (M, g) is a smooth manifold M endowed with a metric g of signature (−, +, . . ., +).Given a non-zero tangent vector 0 timelike or causal) if its tangent vectors ċ(t) are spacelike (lightlike, timelike or causal, respectively), for all t ∈ I.A Lorentzian manifold M is called time-orientable if there exists an everywhere timelike vector field t ∈ Γ(T M ).Such t determines a time-orientation on M .We will denote oriented and time-oriented Lorentzian manifolds by M = (M, g, o, t), where o is the chosen orientation.A timelike or causal curve c : I → M is said to be future directed if g(t, ċ) < 0 and past directed if g(t, ċ) > 0. The chronological future/past I ± M (S) ⊆ M of a subset S ⊆ M consists of all points that can be reached by a future/past directed timelike curve stemming from S. Similarly, the causal future/past J ± M (S) ⊆ M consists of S itself and of all points that can be reached by a future/past directed causal curve stemming from S. By definition, I ± M (S) ⊆ J ± M (S); moreover, recall from e.g.[ONe83, Chapter 14] that e. when all causal curves with endpoints in S lie in S.An example of a causally convex subset is the causally convex hull coincides with the chronological one.(Indeed, any p ∈ J ± M (O) lies along a future/past directed causal curve emanating from some q ∈ O. Since O is open, q can be reached via a future/past directed timelike curve emanating from some r ∈ O.
Consider an oriented and time-oriented globally hyperbolic Lorentzian manifold M ∈ Loc m of dimension m ≥ 2. Let E → M be a real or complex vector bundle of finite rank.Denote the vector space of smooth sections of E by Γ(E) and the vector subspace of compactly supported sections by Γ c (E) ⊆ Γ(E).
Definition 2.8.A Green hyperbolic operator is a linear differential operator P : Γ(E) → Γ(E) that admits retarded and advanced Green's operators G ± , which are linear maps G ± : Γ c (E) → Γ(E) such that, for all ϕ ∈ Γ c (E), the following conditions hold: between the retarded and advanced Green's operators is called the retarded-minus-advanced propagator and their average In [Bar15] it is shown that the retarded and advanced Green's operators associated with a Green hyperbolic operator are unique.
Given a real vector bundle E → M endowed with a fiber metric −, − , i.e. a fiber-wise nondegenerate, symmetric, bilinear form, and denoting the volume form on M by vol M , one defines the integration pairing for all sections ϕ, ϕ ′ ∈ Γ(E) with compact overlapping support, i.e. such that supp(ϕ)∩supp(ϕ ′ ) ⊆ M is compact.Given two vector bundles (E 1 , −, − 1 ), (E 2 , −, − 2 ) endowed with fiber metrics and a linear differential operator ) as the unique linear differential operator such that ) is a formally self-adjoint Green hyperbolic operator, the associated retarded and advanced Green's operators G ± are "formal adjoints" of each other, i.e.

Algebraic QFTs and time-orderable prefactorization algebras
Algebraic quantum field theories (AQFTs) [BFV03, FV15, BSW21] and factorization algebras [CG17, CG21, BPS20] provide two axiomatic frameworks to describe the algebraic structures on the observables of a quantum field theory in various geometric settings.In this subsection we review some basic concepts from these two frameworks in the Lorentzian setting.
We say that two Loc m -morphisms f 1 : M 1 → N ← M 2 : f 2 to a common target are causally disjoint if there exists no causal curve in N connecting their images, i.e.J N (f 1 (M 1 ))∩f 2 (M 2 ) = ∅, where J M (S) := J + M (S) ∪ J − M (S) denotes the union of the causal future and past of a subset Definition 2.9.A Ch C -valued algebraic quantum field theory (AQFT) A on Loc m is a functor A : Loc m → dgAlg C taking values in the category dgAlg C of differential graded algebras that satisfies the following axioms: (i) Einstein causality: For all causally disjoint morphisms f in Ch C commutes, where µ N and µ op N := µ N • γ are the multiplication and the opposite multiplication of A(N ) ∈ dgAlg C ; (ii) Time-slice: For all Cauchy morphisms f : A morphism κ : A → B between AQFTs is a natural transformation.This defines the category AQFT m of Ch C -valued AQFTs as the full subcategory AQFT m ⊆ dgAlg Locm C of the functor category consisting of all functors that satisfy the Einstein causality and time-slice axioms.
Remark 2.10.There exists a more elegant and powerful operadic description [BSW21] of the category AQFT m .This more abstract perspective is particularly useful to endow AQFT m with a model category structure [BSW19], which provides a solid foundation for the study of Ch Cvalued AQFTs.To prove the results of our present paper, we do not have to make explicit use of these techniques.△ Our goal is to construct and compare AQFTs and prefactorization algebras in the Lorentzian setting.For this purpose, we recall below a Lorentzian version of the prefactorization algebras from [CG17], called time-orderable prefactorization algebras [BPS20].This requires some preliminaries.A tuple of Loc m -morphisms (f 1 : When a time-ordering permutation exists, one says that f : M → N in Loc m is time-orderable.(Note that the time-ordering permutation for a tuple may not be unique.For instance, two morphisms f 1 : M 1 → N ← M 2 : f 2 in Loc m are causally disjoint precisely when both (f 1 , f 2 ) and (f 2 , f 1 ) are time-ordered pairs.)A time-orderable 1-tuple (f ) : M → N in Loc m is denoted simply as a morphism f : M → N in Loc m and, for each N ∈ Loc m , we define a unique timeorderable empty tuple ∅ → N .Time-orderable tuples are composable and carry permutation group actions, see [BPS20].These facts are crucial for the next definition.
Definition 2.11.A Ch C -valued time-orderable prefactorization algebra F on Loc m consists of the data listed below: Ch C , called time-ordered product, where F(M ) := n i=1 F(M i ) ∈ Ch C denotes the tensor product.By convention, the time-ordered product assigned to an empty tuple ∅ → N is a morphism C → F(N ) in Ch C from the monoidal unit.These data are subject to the following axioms: (i) For all time-orderable tuples f = (f 1 , . . ., f n ) : M → N and g i = (g i1 , . . ., g ik i ) : (2.28) in Ch C commutes, where γ σ is defined by the symmetric braiding γ of Ch C .
We say that a time-orderable prefactorization algebra F satisfies the time-slice axiom if, for all Cauchy morphism f : that is compatible with the time-ordered products in the sense that, for all time-orderable tuples We denote the category of time-orderable prefactorization algebras on Loc m satisfying the time-slice axiom by tPFA m .

Green's witnesses
In this section we briefly recall the concept of a Green's witness for a complex of linear differential operators, see [BMS22] for more details.This consists of a collection of degree decreasing linear differential operators that enable the explicit construction of retarded and advanced Green's homotopies.The latter are differential graded analogs of the usual retarded and advanced Green's operators, see e.g.[BGP07,Bar15] and also Subsection 2.3, and they will play a key role in our construction of AQFTs and their comparison to time-orderable prefactorization algebras in Section 4. Given a Green's witness, we shall endow the underlying complex of linear differential operators with the following three structures: 1.) a (−1)-shifted Poisson structure τ (−1) , 2.) an unshifted Poisson structure τ (0) and 3.) a symmetric pairing τ D , that we call Dirac pairing, trivializing the (−1)-shifted Poisson structure, i.e. τ (−1) = ∂τ D .We shall show that τ (−1) , τ (0) and τ D are natural when all input data are natural (with respect to the category Loc m of m-dimensional oriented and time-oriented globally hyperbolic Lorentzian manifolds).In particular, we shall construct a functor ( ) whose cochains may be interpreted field-theoretically as linear observables.(Here PoCh R denotes the category whose objects are Poisson cochain complexes (V, τ ), consisting of a cochain complex V ∈ Ch R endowed with an unshifted linear Poisson structure τ , see Definition 2.3, and whose morphisms f : (V, τ ) → (W, ω) are cochain maps f : V → W in Ch R preserving the Poisson structures, i.e. ω • (f ⊗ f ) = τ .)In Theorem 3.13 we shall prove that the functor (F c [1], τ (0) ) satisfies the classical analogs of the Einstein causality and time-slice axioms.
3.1 τ (−1) , τ (0) and τ D over a fixed globally hyperbolic Lorentzian manifold Given a (Z-)graded (R-)vector bundle F → M (degree-wise of finite rank) over an oriented and time-oriented globally hyperbolic Lorentzian manifold M ∈ Loc m , we denote by the vector space of degree n smooth sections, i.e. the smooth sections of the degree n vector bundle F n → M , and by the vector space of degree n smooth sections with compact support.
Definition 3.1.A complex of linear differential operators (F, Q) over M ∈ Loc m consists of a graded vector bundle F → M and of a collection ) n∈Z of degree increasing linear differential operators such that Q n+1 Q n = 0, for all n ∈ Z.We denote by F(M ) ∈ Ch R the cochain complex of sections associated with the complex of linear differential operators (F, Q).
Remark 3.2.The compatibility condition (3.3) implies that the integration pairing defined by for all ψ ∈ F c (M ) and ϕ ∈ F(M ), is a cochain map.△ Definition 3.3.A (formally self-adjoint) Green's witness W = (W n ) n∈Z for a complex of linear differential operators (F, Q) endowed with a compatible (−1)-shifted fiber metric (−, −) consists of a collection of degree decreasing linear differential operators W n : F(M ) n → F(M ) n−1 such that the following conditions hold: (i) For all n ∈ Z, for all homogeneous sections ϕ 1 , ϕ 2 ∈ F(M ) with compact overlapping support.
Remark 3.4.Some direct consequences of Definition 3.3 are listed below: (1) For all n ∈ Z, there exist unique retarded and advanced Green's operators G n ± : F c (M ) n → F(M ) n associated with the Green hyperbolic operators P n ; (2) It follows that P W = W P and P Q = Q P , hence also (3) P is formally self-adjoint, i.e.M (P ϕ 1 , ϕ 2 ) vol M = M (ϕ 1 , P ϕ 2 ) vol M , for all sections ϕ 1 , ϕ 2 ∈ F(M ) with compact overlapping support.It follows that , for all sections ψ 1 , ψ 2 ∈ F c (M ) with compact support, and hence also Example 3.6.Our first example of a free BV theory over M ∈ Loc m is obtained from an ordinary field theory, which is defined by a formally self-adjoint Green hyperbolic operator P acting on sections of a vector bundle E over M endowed with a fiber metric −, − .To these data one assigns the free BV theory (F P , Q P , (−, −) P , W P ) consisting of the complex of linear differential operators (F P , Q P ) := E P / / E (3.5a) concentrated in degrees 0 and 1, of the compatible (−1)-shifted fiber metric (−, −) P uniquely determined by for all ϕ ∈ F 0 P = E and ϕ ‡ ∈ F 1 P = E over the same base point, and of the Green's witness Here and in the following examples we decided to use a convenient graphical visualization for a Green's witness as a sequence of linear differential operators, which is pointing from right to left because W decreases the degree.It is important to emphasize that this sequence is in general not a chain complex because a Green's witness is not necessarily square-zero.▽ Example 3.7.The free BV theory (F CS , Q CS , (−, −) CS , W CS ) associated with linear Chern-Simons theory on M ∈ Loc 3 consists of the complex of linear differential operators concentrated between degrees −1 and 2 (this is the 1-shift of the de Rham complex up to a global sign), of the (−1)-shifted fiber metric (−, −) CS uniquely determined by over the same base point, where ∧ denotes the wedge product on differential forms and * denotes the Hodge operator on M , and of the Green's witness concentrated between degrees −p and p + 1, of the (−1)-shifted fiber metric (−, −) MW uniquely determined by for all k = 0, . . ., p, a ∈ F −k MW = Λ p−k M and a ‡ ∈ F k+1 MW = Λ p−k M over the same base point, where s 1 := 1 and s k := (−1) k s k−1 , for k = 2, . . ., p + 1, and of the Green's witness Note that for p = 1 Maxwell p-forms recover linear Yang-Mills theory.▽ Let (F, Q, (−, −), W ) be free BV theory.We define the retarded/advanced Green's homotopy where G ± denotes the retarded/advanced Green's operator associated with P , see Definition 3.3 and Remark 3.4.(In (3.8) we used Remark 3.4 (2) and that W preserves supports.)Note that the retarded/advanced Green's homotopy Λ ± ∈ [F c (M ), F(M )] −1 is a cochain homotopy that trivializes the cochain map j : F c (M ) → F(M ) in Ch R forgetting compact supports.More explicitly, one computes where in the first step we used the definition of the internal hom differential ∂, the second step follows from Remark 3.4 (2) and in the last step we used that G ± is the retarded/advanced Green's operator associated with P .
Remark 3.9.Λ ± as defined in (3.8) is a specific choice of a retarded/advanced Green's homotopy in the more general sense of [BMS22, Definition 3.5].Such level of generality plays a crucial role to ensure uniqueness of retarded/advanced Green's homotopies, see [BMS22, Proposition 3.9].This general and more abstract concept of a retarded/advanced Green's homotopy is not needed for the present paper because a Green's witness W for the complex of linear differential operators (F, Q) is given, which allows us to consider the explicit choices Λ ± from (3.8).This considerably simplifies our analysis, in particular in view of naturality with respect to M ∈ Loc m , see Subsection 3.2 below.△ We shall now endow the complex F c (M )[1] ∈ Ch R of linear observables with both a (−1)shifted Poisson structure τ (−1) and an unshifted one τ (0) .Furthermore, we shall construct a symmetric pairing τ D , called Dirac pairing, that trivializes τ (−1) , i.e. ∂τ D = τ (−1) .The key ingredients for our construction are the integration pairing ((−, −)) from (3.4) and the retarded and advanced Green's homotopies Λ ± from (3.8).By taking their difference, we define the retarded-minus-advanced cochain map in Ch R , where Λ ± are regarded here as 0-cochains in given by (−1) n in degree n).Note that Λ is a cochain map because ∂Λ ± = j.Similarly, we define the Dirac homotopy as a graded linear map of degree 0. We have seen in (3.9) that the cochain map j : F c (M ) → F(M ) is trivialized by Λ ± .It follows that a similar result is achieved by the Dirac homotopy Λ D , namely First, we define the (−1)-shifted Poisson structure in Ch R , where γ denotes the symmetric braiding.To confirm that (3.13) defines a (−1)-shifted Poisson structure we have to check symmetry τ (−1) • γ = τ (−1) .Indeed, for all homogeneous sections ψ 1 , ψ 2 ∈ F c (M ) [1] with compact support, one has where in the first and last steps we used the definition of τ (−1) from (3.13) and in the second step we used that the fiber metric (−, −) is graded anti-symmetric, see Definition 3.1.
Second, we define the unshifted Poisson structure in Ch R .To confirm that (3.15) defines an unshifted Poisson structure we have to check antisymmetry τ (0) • γ = −τ (0) .Indeed, for all homogeneous sections ψ 1 , ψ 2 ∈ F c (M ) [1] with compact support, one has where in the first and last steps we used the definition of τ (0) from (3.15), in the second step we used that the fiber metric (−, −) is graded anti-symmetric, see Definition 3.1, and in the third step we used Definition 3.3 (iii) and Remark 3.4 (3).
Finally, we define the Dirac pairing (3.18) Indeed, for all homogeneous sections ψ 1 , ψ 2 ∈ F c (M )[1] with compact support, one has where in the first step we used the definition of τ D from (3.17), in the second step we used (3.3), in the third step we used (3.12) and in the last step we used the definition of τ (−1) from (3.13).Definition 3.10.A natural collection of free BV theories (F M , Q M , (−, −) M , W M ) M ∈Locm consists of natural vector bundles F n , natural linear differential operators

Properties of τ
) is a free BV theory in the sense of Definition 3.5.
The concepts of natural vector bundles, natural fiber metrics and natural differential operators, which are relevant for the definition above, are recalled in Appendix A.
Example 3.11.Let us upgrade Examples 3.6, 3.7 and 3.8 to natural collections of free BV theories as formalized in Definition 3.10.
Concerning the natural upgrade of Example 3.6, it suffices to take as input a natural vector bundle E endowed with a natural fiber metric −, − and a natural linear differential operator P defined on E, whose components P M are formally self-adjoint Green hyperbolic operators, for all M ∈ Loc m .Taking the natural Green's witness given, for all M ∈ Loc m , by the identity as in Example 3.6, one obtains a natural collection of free BV theories.For instance, the natural collection of free BV theories associated with the real Klein-Gordon field of mass m ≥ 0 is obtained by taking E = R to be the natural trivial line bundle, whose components are the trivial line bundles M × R → M , endowed with its canonical natural fiber metric −, − , given component-wise by the multiplication on R, and the Klein-Gordon operator P = + m 2 , whose naturality follows from the fact that morphisms of Loc m are isometries.
The upgrade of Examples 3.7 and 3.8 to natural collections of free BV theories is obtained as follows.First, consider the natural vector bundles of differential k-forms Λ k , whose components are the vector bundles Λ k M → M .(Naturality follows from the fact that morphisms of Loc m are open embeddings.)Second, note that the wedge product ∧ of differential forms, the Hodge operator * and the de Rham differential d are natural with respect to the morphisms in Loc m .(This relies also on the fact that morphisms of Loc m are orientation-preserving isometries.)This defines a natural structure in the sense of Definition 3.10 on the collection of free BV theories from Examples 3.7 and 3.8, which describe linear Chern-Simons theory and Maxwell p-forms.▽ We summarize below the key facts that will play a crucial role in the rest of the paper.These are part of Definition 3.10, or follow from it and the constructions outlined in Appendix A, especially (A.1), (A.5), (A.7) and (A.8).For all f : M → N in Loc m , one has the following: (1) A pushforward cochain map f * : F c (M ) → F c (N ) in Ch R for compactly supported sections and a pullback cochain map f * : F(N ) → F(M ) in Ch R for sections.(In particular, (2) Naturality of the integration pairing (3.4), i.e. the diagram in Ch R commutes.
(1) and (3) entail that also the Green hyperbolic operators Finally, for all M ∈ Loc m , let us consider the (−1)-shifted Poisson structures τ M (−1) from (3.13), the unshifted Poisson structures τ M (0) from (3.15) and the Dirac pairings τ M D from (3.17), with additional superscripts emphasizing the underlying object in Loc m .As a consequence of (3.21) and of the naturality of the integration pairing, see (3.20), one obtains the following result.Lemma 3.12.For all f : M → N in Loc m , the following holds Proof.The first equality follows immediately from (3.13) and (3.20).To prove also the second equality, recall (3.15) and, for all where we used (3.20) in the first and (3.21) in the second step.Recalling (3.17), the proof of the third equality is the same.
This means that τ M (−1) , τ M (0) and τ M D are the components at M ∈ Loc m of the natural transformations τ (−1) , τ (0) and τ D , respectively.In particular, the assignment to each object .)The next result shows that classical analogs of the Einstein causality and time-slice axioms hold.To simplify our notation, from now on we shall suppress the superscripts and subscripts emphasizing the underlying object of Loc m , whenever this information can be inferred from the context.Theorem 3.13.Let (F M , Q M , (−, −) M , W M ) M ∈Locm be a natural collection of free BV theories.
(b) For all Cauchy morphisms f : M → N in Loc m , the pushforward cochain map Proof.Item (a) follows from J N (f 1 (M 1 )) ∩ f 2 (M 2 ) = ∅ (because f 1 and f 2 are causally disjoint), the definition of the unshifted Poisson structure τ (0) and the support properties of retarded and advanced Green's operators, see (3.15) and Definition 2.8.
To prove also item (b), we shall construct a quasi-inverse g : Quasi-inverse g: We construct a candidate quasi-inverse as the (unique) cochain map in Ch R that satisfies the equation lies in the image of the degree-wise injective cochain map j f * .Indeed, the support of the section −(∂(χ (The latter subset is compact by [BGP07, Corollary A.5.4] and contained in f (M ) because by construction J Homotopy η: We construct a candidate homotopy as the (unique) (−1)-cochain that satisfies the equation where Λ ± are regarded here as 0-cochains in given by (−1) n in degree n).Such (−1)-cochain η exists (uniquely) because, for all sections ψ ∈ F c (N )[1], the section χ ∓ Λ ± ψ ∈ F(N )[1] lies in the image of the degree-wise injective cochain map j that forgets compact supports.Indeed, the support of the section where in the first step we used that j is a cochain map and the equation defining η, in the second step we used the Leibniz rule of ∂ with respect to the composition, χ + + χ − = 1 (hence ∂χ + = −∂χ − ), ∂Λ ± = j and Λ = Λ + − Λ − and in the last step we used ∂Λ = 0 and (3.26).

Homotopy ζ:
We construct a candidate homotopy as the (unique that satisfies the equation Such (−1)-cochain ζ exists (uniquely) because, for all homogeneous sections ψ ∈ F c (M )[1], the section χ ∓ Λ ± f * ψ ∈ F(N )[1] lies in the image of the degree-wise injective cochain map f * .Indeed, the support of the section where in the first step we used that f * is a cochain map and the definition of ζ and in the last step we used ∂η = id − f * g.
To conclude this section, we record a simple result relating τ (0) and τ D via time-ordering.
M ∈Locm be a natural collection of free BV theories.Then, for all time-ordered pairs recalling the support properties of retarded and advanced Green's operators from Definition 2.8, one computes The first step uses the definition of the unshifted Poisson structure τ (0) , see (3.15).Both the second and third steps use that The last step uses the definition of the Dirac pairing τ D , see (3.17).

Quantizations and comparison
In this section we shall present two a priori different approaches to the quantization of a natural collection (F M , Q M , (−, −) M , W M ) M ∈Locm of free BV theories.First, in Subsection 4.1 we shall construct a time-orderable prefactorization algebra F ∈ tPFA m by deforming the ordinary differential of the symmetric algebra Sym(F c (M )[1]) generated by linear observables with the BV Laplacian, as prescribed by the BV formalism [CG17,CG21].Second, in Subsection 4.2 we shall construct an AQFT A ∈ AQFT m by deforming the commutative multiplication of Sym(F c (M )[1]) ∈ dgAlg C to the non-commutative Moyal-Weyl star product.These two constructions involve different input data.More specifically, the time-orderable prefactorization algebra F ∈ tPFA m relies only on the natural (−1)-shifted fiber metric (−, −) through the natural (−1)-shifted Poisson structure τ (−1) (except for the time-slice axiom), while the AQFT A ∈ AQFT m relies also on the natural Green's witness W through the natural unshifted Poisson structure τ (0) .Last, we shall show in Subsection 4.3 that, when both (−, −) and W are given, the natural Dirac pairing τ D leads to an isomorphism T : F → F A in tPFA m to the time-orderable prefactorization algebra F A ∈ tPFA m canonically associated with A ∈ AQFT m , see [BPS20].Let us mention that the deformation parameter > 0 will not be formal in our constructions below.Indeed, all expansions in powers of that appear later on actually stop at finite order, see for instance the comment after (4.13).
As a preparatory step, let us present a geometric construction that will be used frequently in the rest of the paper.
Proof.Recalling Subsection 2.3, we define the subset as the causally convex hull of the union of the images of f i , for i = 1, . . ., n − 1.Since the images are open, M ⊆ N is open and causally convex.Endowing it with the restriction of the orientation, time-orientation and metric of N defines an object M ∈ Loc m and promotes the subset inclusion By contraposition, suppose that the intersection is not empty.Then there exists a future directed causal curve in N emanating from f (M ) and reaching f n (M n ).Since any point in the causally convex hull M is by definition in the causal future of f i (M i ), for some i = 1, . . ., n − 1, it follows that there exists a future directed causal curve in N emanating from f i (M i ), for some i = 1, . . ., n − 1, and reaching f n (M n ), leading to a contradiction with the hypothesis that f is time-ordered.

BV quantization
Let (F M , Q M , (−, −) M , W M ) M ∈Locm be a natural collection of free BV theories.Consider the symmetric algebra Sym(F c (M )[1]) ∈ dgCAlg C generated by the complexification of [1] and the graded Leibniz rule.BV quantization consists of deforming Q by means of the BV Laplacian which is the Laplacian associated to the (−1)-shifted Poisson structure τ (−1) , see Definition 2.6 and (3.13).Explicitly, one defines the degree increasing graded linear map where > 0 is Planck's constant and i ∈ C is the imaginary unit.Note that Q defines a new differential since it squares to zero where we used Q 2 = 0, ∂∆ BV = ∆ ∂τ (−1) = 0 and ∆ 2 BV = −∆ 2 BV = 0, see (2.16) and (2.17).We define the cochain complex of quantum observables by replacing the original differential Q with the quantized one Q .The assignment Loc m ∋ M → F(M ) ∈ Ch C of the cochain complex of quantum observables can be promoted to a timeorderable prefactorization algebra F ∈ tPFA m .For this purpose, we need to define time-ordered products that are compatible with the quantized differential Q .This is the goal of the next proposition.
Proposition 4.2.Let f : M → N be a time-orderable tuple in Loc m of length n.Then the time-ordered product Here f i * denotes the symmetric algebra extension of the pushforward cochain map for compactly supported sections, see Subsection 3.2, and µ (n) denotes the n-ary multiplication on the symmetric algebra Hence, it suffices to prove the analog ∆ BV F(f ) = F(f ) ∆ BV ⊗ for the BV Laplacian ∆ BV .Furthermore, since the symmetric algebra multiplication µ is commutative, it suffices to prove the claim for f time-ordered.We argue by induction on the length n.For n = 0, the time-ordered product C → F(N ) defined above assigns the unit µ (0) = ½ of the symmetric algebra, hence the claim follows from ∆ BV (½) = 0, see Definition 2.6.For n = 1, F(f ) = f * , hence the claim follows because the BV Laplacian ∆ BV inherits the naturality of the (−1)-shifted Poisson structure τ (−1) , see (2.18) and (3.22).For n = 2, one computes where in the first and last steps we used the definition of the time-ordered product F(f 1 , f 2 ), in the second step we used the degree increasing graded endomorphism {{−, −}} (−1) := {{−, −}} τ (−1) to spell out the modified Leibniz rule of ∆ BV , see Definitions 2.4 and 2.6, and in the third step we used naturality of the BV Laplacian ∆ BV and that {{−, −}} (−1) vanishes on the image of where in the first and last steps we used the definition of the time-ordered product F(f ) and in the second step we used µ .This shows that the filtration is complete, see [EM62].Furthermore, for p ≥ 0, the p-th component of the associated graded cochain complex is isomorphic to the p-th symmetric power of F c (L)[1] ∈ Ch C (endowed with the original differential Q) because the BV Laplacian ∆ BV lowers the symmetric power by 2, see (2.14).Functoriality with respect to L ∈ Loc m of the filtration (4.10) and naturality of the isomorphism (4.11) entail that, for all f : M → N in Loc m , the diagram   The Moyal-Weyl star product µ is manifestly a degree preserving graded linear map.Furthermore, it is associative and unital with respect to ½ ∈ Sym(F c (M )[1]) as a consequence of the properties of the degree preserving graded endomorphism {{−, −}} (0) = {{−, −}} τ (0) and of the exponential.Let us also check that the Moyal-Weyl star product µ is compatible with the where in the first step we used the compatibility ∂µ = 0 of the symmetric algebra multiplication µ with the differential Q, in the second step we expanded the exponential series and applied the Leibniz rule for ∂ and in the last step we used that ∂{{−, −}} (0) = {{−, −}} ∂τ (0) = 0 vanishes, see (2.9) and recall that τ (0) is a cochain map.Therefore, we define the quantized differential graded algebra (4.17) To promote the assignment Loc m ∋ M → A(M ) ∈ dgAlg C of the quantized differential graded algebra to a functor, we check the naturality of the Moyal-Weyl star product µ with respect to morphisms f : M → N in Loc m , i.e.
where in the first step we used naturality of the symmetric algebra multiplication µ and in the second step we used the naturality of the unshifted Poisson structure τ (0) , see (3.22), in combination with (2.10) at all orders in .We are now ready to define the functor that assigns to any object M ∈ Loc m the differential graded algebra A(M ) ∈ dgAlg C and to any morphism f : M → N in Loc m the morphism A(f ) : A(M ) → A(N ) in dgAlg C , whose underlying cochain map is the symmetric algebra extension of the pushforward cochain maps with the symmetric algebra multiplication µ.Since the latter is commutative, the Einstein causality axiom follows.
Second, let us check the time-slice axiom.Given a Cauchy morphism f : M → N in Loc m , it suffices to show that the cochain map underlying A(f ) : A(M ) → A(N ) in dgAlg C is a quasiisomorphism.This is the case because the cochain map underlying A(f ) is by definition the symmetric algebra extension of the pushforward cochain map f which is a quasi-isomorphism by Theorem 3.13.
Example 4.7.Taking the natural free BV theories from Example 3.11 as inputs, the constructions and results of this subsection produce AQFTs that quantize ordinary field theories, linear Chern-Simons theory and Maxwell p-forms (including linear Yang-Mills theory for p = 1).In the case of the Klein-Gordon field and of Maxwell p-forms, earlier constructions of the same AQFTs can be found in [BBS20,AB22].▽

Comparison
This subsection compares the two different quantization schemes from Sections 4.1 and 4.2.More specifically, we establish an isomorphism between the time-orderable prefactorization algebra F ∈ tPFA m constructed using the BV formalism and the time-orderable prefactorization algebra F A ∈ tPFA m canonically associated with the AQFT A ∈ AQFT m .For this purpose, recall from [BPS20] that F A ∈ tPFA m consists of the data listed below: (a) For each M ∈ Loc m , the cochain complex  The above data fulfill the axioms of Definition 2.11, see [BPS20] for more details.
In preparation for our comparison result stated in Theorem 4.9, the next lemma explains how the time-orderable prefactorization algebra F A ∈ tPFA m captures the usual time-ordered products built out of the Dirac multiplication  Then one computes where in the first step we used the definition of the Dirac multiplication µ D , see (4.23), in the second step we used (4.25) and in the last step we used the definition of the time-ordered product Proof.Suppose that T as defined above is a morphism of time-orderable prefactorization algebras.Then it is also an isomorphism with inverse T −1 := exp(− i ∆ D ) : F A → F in tPFA m .Therefore, it suffices to check that T is a morphism of time-orderable prefactorization algebras.We split this check in two parts.First, we show the compatibility with differentials, i.e. that, for all M ∈ Loc m , the M -component In the first step we used the definition of ∂.In the second step we expanded the exponential that defines T M (recall that the series evaluated on any a ∈ Sym(F c (M )[1]) truncates to a finite sum) and used that ∂ is linear and vanishes on id.In the last step we used the Leibniz rule for ∂ with respect to composition, (4.29) and the definition of T M .Equation (4.30) means that Q • T M = T M • Q , which shows that T M : F(M ) → F A (M ) is a cochain map.
Compatibility with time-ordered products: Since F, F A ∈ tPFA m are time-orderable prefactorization algebras, their time-ordered products are equivariant with respect to permutations, see Definition 2.11.Hence, it suffices to show the compatibility of T with time-ordered products for f : M → N time-ordered.We argue by induction on the length n of f .For n = 0 this is trivial.For n = 1, the claim follows from naturality of the Dirac Laplacian ∆ D • f * = f * • ∆ D , see (2.18) and (3.22).For n = 2, one computes where in the first step we used T N • µ = µ D • (T N ⊗ T N ), which follows from (2.15), and in the last step we used the claim for n = 1 and Lemma 4.
where in the first and last steps we used the composition and identity axioms of time-orderable prefactorization algebras and f • f ′ i = f i , for all i = 1, . . ., n − 1, and in the second step we used the claim for lengths 2 and n − 1.
Example 4.10.The abstract comparison result established in Theorem 4.9, when specialized to the time-orderable prefactorization algebras from Example 4.4 and the corresponding AQFTs from Example 4.7, provides concrete comparison results between the time-orderable prefactorization algebras and the AQFTs quantizing ordinary field theories, linear Chern-Simons theory and Maxwell p-forms (including linear Yang-Mills theory for p = 1).Our result generalizes the earlier comparison result in [GR20], which is formulated only for ordinary field theories, to the case of gauge and also higher gauge theories.▽ by means of the BV Laplacian ∆ BV defined from the (−1)-shifted Poisson structure τ (−1) ; Subsection 4.2 quantizes (F M , Q M , (−, −) M , W M ) M ∈Locm as an AQFT A ∈ AQFT m by deforming the original commutative multiplication µ of the symmetric algebra Sym(F c (M )[1]) ∈ dgCAlg C to the (in general non-commutative) Moyal-Weyl star product µ by means of the bi-derivation {{−, −}} (0) extending the unshifted Poisson structure τ (0) ; Subsection 4.3 concludes the paper with constructing in Theorem 4.9 an isomorphism T := exp( i ∆ D ) : F → F A in tPFA m , where ∆ D denotes the Dirac Laplacian defined from the Dirac pairing τ D .
20) of a subset S ⊆ M , i.e. the smallest causally convex subset of M that contains S. Definition 2.7.An oriented and time-oriented Lorentzian manifold M is called globally hyperbolic if it admits a Cauchy surface Σ ⊂ M , i.e. a subset that is met exactly once by any inextendible future directed timelike curve in M .Loc m denotes the category whose objects are all m-dimensional oriented and time-oriented globally hyperbolic Lorentzian manifolds M and whose morphisms are all orientation and time-orientation preserving isometric embeddings f : M → M ′ with open and causally convex image f (M ) ⊆ M ′ .For M ∈ Loc m and O ⊆ M open, one has that the causal future/past denote respectively the retarded-minusadvanced and Dirac propagators.These observations will be used frequently in our constructions in this paper.△ In analogy with the Riemannian setting [CG21], we introduce the following terminology.Definition 3.5.A free BV theory (F, Q, (−, −), W ) on M ∈ Loc m consists of a complex of linear differential operators (F, Q) with a compatible (−1)-shifted fiber metric (−, −) and a Green's witness W . Several examples of free BV theories, closely related to the examples from [BBS20, AB22, BMS22], are presented below.
(−1) , τ (0) and τ D Let us now consider a collection (F M , Q M , (−, −) M , W M ) M ∈Locm of free BV theories, indexed by M ∈ Loc m .We assume that (F M , Q M , (−, −) M , W M ) M ∈Locm is natural with respect to the morphisms f : M → N in Loc m in the sense of the next definition.
natural, hence P N f * = f * P M and P M f * = f * P N , for all f : M → N in Loc m .As a consequence of the naturality of P = (P M ) M ∈Locm and of the theory of Green hyperbolic operators, for all f : M → N in Loc m , one has the usual naturality property f * G N ± f * = G M ± for the retarded/advanced Green's operator G M ± associated with P M , see [BG11], as well as the analogs f * G N f * = G M and and f * G N D f * = G M D for the retarded-minus-advanced propagator G M := G M + − G M − and for the Dirac propagator G M D := 1 2 (G M + + G M − ).Therefore, the retarded/advanced Green's homotopies Λ M ± := W M G M ± , the retarded-minus advanced cochain maps Λ M := Λ M + − Λ M − and the Dirac homotopies Λ M D := 1 2 (Λ M + + Λ M − ) inherit the same naturality, that is, for all Loc m -morphisms f : M → N , one has R .(Note that (3.22) expresses the necessary compatibility of f * with the unshifted Poisson structures τ M (0) and τ N (0) by hypothesis a Cauchy morphism, let us consider two spacelike Cauchy surfaces Σ ± ⊂ N lying inside the image of f such that Σ + ⊂ I + N (Σ − ) is contained in the chronological future of Σ − .Choose a partition of unity {χ + , χ − } subordinate to the open cover {I + N (Σ − ), I − N (Σ + )} of N .

Lemma 4. 1 .
Let f = (f 1 , . . ., f n ) : M = (M 1 , . . ., M n ) → N be a time-ordered tuple in Loc m of length n ≥ 2. Then there exist M ∈ Loc m , f : M → N in Loc m and a time-ordered tuple f and τ (−1) vanishes on sections with disjoint supports, see (3.13).For n ≥ 3, taking M ∈ Loc m , f : M → N in Loc m and a time-ordered tuple f ′ : (M 1 , . . ., M n−1 ) → M in Loc m as provided by Lemma 4.1, one computes and the naturality of the symmetric algebra multiplication µ.Hence, the claim for length n ≥ 3 follows from lengths 2 and n − 1.With these preparations, we define the time-orderable prefactorization algebra F ∈ tPFA m by the data listed below:(a) For each M ∈ Loc m , the cochain complex F(M ) ∈ Ch C from (4.6);(b) For each time-orderable tuple f : M → N in Loc m , the time-ordered product F(f ) :F(M ) → F(N ) in Ch C from Proposition 4.2.Note that these data satisfy the axioms of Definition 2.11 because F c [1] : Loc m → Ch R is a functor, see Subsection 3.2, and the symmetric algebra multiplication µ is associative, unital and commutative.The resulting time-orderable prefactorization algebra F ∈ tPFA m satisfies the time-slice axiom, as explained by the next proposition.Proposition 4.3.If f : M → N in Loc m is a Cauchy morphism, then F(f ) : F(M ) → F(N ) in Ch C is a quasi-isomorphism.Proof.For any L ∈ Loc m , consider the filtration ofF(L) = n≥0 Sym n (F c (L)[1]) ∈ Ch C associated with symmetric powers.Explicitly, we denote the subcomplex of F(L) consisting of symmetric powers up to p ≥ 0 byF p (F(L)) := p n=0 Sym n (F c (L)[1]), Q ⊆ F(L) .(4.10)(Note that this filtration is compatible with the quantized differential Q = Q + i ∆ BV because the original differential Q preserves the symmetric power and the BV Laplacian ∆ BV lowers the symmetric power by 2.) The resulting filtration is bounded from below, i.e.F p (F(L)) = 0 vanishes, for all p < 0. The quotient maps F(L) → F(L)/F p (F(L)) in Ch C , for all p ∈ Z, form a universal cone, i.e.F(L) ∼ = lim p∈Z F(L)/F p (F(L)) Let (F M , Q M , (−, −) M , W M ) M ∈Locm be a natural collection of free BV theories and consider again the symmetric algebra Sym(F c (M )[1]) ∈ dgCAlg C generated by the complexification of F c (M )[1] ∈ Ch R .Canonical quantization can be realized by deforming the commutative multiplication µ of the symmetric algebra Sym(F c (M )[1]) ∈ dgCAlg C to the Moyal-Weyl star product .13) where {{−, −}} (0) := {{−, −}} τ (0) is the degree preserving graded endomorphism associated with the unshifted Poisson structure τ (0) , see Definition 2.4 and (3.15).Note that, for all polynomials a, b ∈ Sym(F c (M )[1]), the exponential series defining µ (a ⊗ b) truncates to a finite sum.In particular, there is no need to regard as a formal parameter.Remark 4.5.The Moyal-Weyl star product µ is a non-commutative deformation of the commutative multiplication µ of the symmetric algebra Sym(F c (M )[1]) ∈ dgCAlg C in the sense that the multiplications µ = µ + O( ) (4.14) coincide up to terms of order at least and, moreover, the µ -commutator [−, −] = i {−, −} (0) + O( 2 ) (4.15) is proportional to the Poisson bracket {−, −} (0) := µ • {{−, −}} (0) , see Remark 2.5, up to terms of order at least 2 .△ 22) in Ch C , where n denotes the length of the tuple f , ρ is a time-ordering permutation for f and µ (ρ) := µ (n) • γ ρ denotes the n-ary Moyal-Weyl star product in the order prescribed by ρ.(The Einstein causality axiom of A ensures that F A (f ) does not depend on the choice of the time-ordering permutation ρ.) .23) where {{−, −}} D := {{−, −}} τ D denotes the degree preserving graded endomorphism associated with the natural Dirac pairing τ D (3.17).Note that the Dirac multiplication µ D is associative, unital with respect to ½ ∈ Sym(F c (M )[1]) and commutative because the Dirac pairing τ D is symmetric; however, it is not compatible with the differential Q of F A (M ) ∈ Ch C because ∂τ D = τ (−1) does not vanish, see (2.9) and (3.18).Furthermore, the naturality of τ D and that of the symmetric algebra multiplication µ entail that the Dirac multiplication µ D is natural too.

FfD
A (f 1 , f 2 ), see (4.22).For n ≥ 3, taking M ∈ Loc m , f : M → N in Loc m and a time-ordered tuple f ′ : (M 1 , . . ., M n−1 ) → M in Loc m as provided by Lemma 4.1, one computesµ i * = µ D • (f * ⊗ f n * ) • µ = µ D • (µ (n−1) D ⊗ id), f • f ′ i = f i, for all i = 1, . . ., n − 1, and the naturality of the Dirac multiplication µ D .Hence, the claim for length n ≥ 3 follows from lengths 2 and n − 1.The alternative description from Lemma 4.8 of the time-ordered products of F A ∈ tPFA m plays a key role in the proof of our main result.Theorem 4.9.The time-orderable prefactorization algebra F ∈ tPFA m (constructed via the BV formalism in Subsection 4.1) and the time-orderable prefactorization algebra F A ∈ tPFA m associated with the AQFT A ∈ AQFT m (constructed via the Moyal-Weyl star product in Subsection 4.2) are isomorphic.Explicitly, the time-ordering mapT := exp( i ∆ D ) : F ∼ = −→ F A (4.28)in tPFA m is an isomorphism.Here ∆ D := ∆ τ D , called Dirac Laplacian, is the Laplacian associated with the Dirac pairing τ D , see Definition 2.6 and (3.17).
▽ 4.2 Moyal-Weyl star product The next proposition shows that A is an AQFT.The functor A : Loc m → dgAlg C from (4.19) satisfies the Einstein causality and time-slice axioms of Definition 2.9, hence A ∈ AQFT m is an AQFT.Proof.First, let us check the Einstein causality axiom.For causally disjoint morphisms f 1 :M 1 → N ← M 2 : f 2 in Loc m ,Definition 2.4 applied to the unshifted Poisson structure τ (0) and