A skeletal model for 2d conformal AQFTs

A simple model for the localization of the category $\mathbf{CLoc}_2$ of oriented and time-oriented globally hyperbolic conformal Lorentzian $2$-manifolds at all Cauchy morphisms is constructed. This provides an equivalent description of $2$-dimensional conformal algebraic quantum field theories (AQFTs) satisfying the time-slice axiom in terms of only two algebras, one for the $2$-dimensional Minkowski spacetime and one for the flat cylinder, together with a suitable action of two copies of the orientation preserving embeddings of oriented $1$-manifolds. The latter result is used to construct adjunctions between the categories of $2$-dimensional and chiral conformal AQFTs whose right adjoints formalize and generalize Rehren's chiral observables.


Introduction and summary
This work is a categorical study of 2-dimensional conformal field theories from the perspective of algebraic quantum field theory (AQFT). To better illustrate our main results and their significance, let us recall the broader context. 2-dimensional conformal AQFTs come essentially in two different flavors, which are often called chiral and full. Loosely speaking, a chiral theory is sensitive to only one of the two light-cone coordinates and thus can be formalized in terms of a net of algebras on a single light ray R, or on its compactification given by the circle T = R/Z. We refer the reader to e.g. [KL04,Kaw15] for the precise axiomatic framework for chiral conformal AQFTs and also to [BDH15] for a natural coordinate-free formulation. In contrast to this, a full theory is sensitive to both light-cone coordinates and formalized in terms of a net of algebras on the Minkowski spacetime M, or on its conformal compactification given by the flat cylinder M /Z. See e.g. the review article [Reh15] for more details. Such full conformal AQFTs admit an interesting generalization in the spirit of locally covariant AQFT [BFV03,FV15], which was studied first by Pinamonti [Pin09]. This generalization treats all conformal spacetimes on the same footing and formalizes a theory in terms of a functor A : CLoc 2 → Alg from the category CLoc 2 of oriented and time-oriented globally hyperbolic conformal Lorentzian 2-manifolds to a suitable category Alg of algebras. (The choice of the category Alg depends on the context. Popular choices are the category of associative and unital * -algebras, the category of C * -algebras or the category of von Neumann algebras. Our main results in this paper hold true for all these cases and many more, see Definition 2.1.) Such functor has to satisfy certain axioms, most notably Einstein causality and the time-slice axiom. Our work is developed in the locally covariant framework for 2-dimensional conformal AQFTs.
The traditional description of a theory in terms of a functor A : CLoc 2 → Alg satisfying Einstein causality and the time-slice axiom is far from being efficient: To each of the infinitely many (isomorphism classes of) objects M ∈ CLoc 2 one has to assign an algebra A(M ) ∈ Alg in a functorial way such that Einstein causality and the time-slice axiom hold true. Note that such assignment is to a large extent redundant, in particular due to the time-slice axiom that demands the algebra map A(f ) : A(M ) → A(M ′ ) to be an isomorphism for every Cauchy morphism f : M → M ′ , i.e. for every (orientation and time-orientation preserving) conformal embedding whose (causally convex) image f (M ) ⊆ M ′ contains a Cauchy surface of M ′ . One of the main results of this paper is Theorem 4.1 which provides an equivalent description of 2-dimensional conformal AQFTs that strips off all the redundant data; we call this the skeletal model. Our skeletal description of a 2-dimensional conformal AQFT is indeed much more efficient than the traditional one as it consists of only two algebras, one for the 2-dimensional Minkowski spacetime M and one for the flat cylinder M /Z, together with a suitable action of two copies of the orientation preserving embeddings of oriented 1-manifolds. It is important to emphasize that our result is purely categorical and hence it does not rely on specific analytical features of the algebras under consideration. In particular, it also holds true for associative and unital * -algebras that carry no topology at all.
The method of proof for Theorem 4.1 involves working in the very broad setting of AQFTs on orthogonal categories [BSW21], which are general types of spacetime categories that carry information about independent pairs of subsystems, see Definitions 2.3 and 2.5. Implementing the time-slice axiom can be achieved in this framework by a localization of orthogonal categories, see Proposition 2.12. The key result leading to our skeletal model for 2-dimensional conformal AQFTs is our explicit and very simple description in Theorem 3.9 of the localization of the category CLoc 2 at all Cauchy morphisms, which crucially relies on embedding theorems [FM16,Mon15] from 2-dimensional conformal Lorentzian geometry. Hence, our skeletal model seems to be a specific feature of 2-dimensional conformal AQFTs that is not likely to generalize to higher dimensions.
Besides obtaining deep insights into the algebraic structure of 2-dimensional conformal AQFTs, our skeletal model is also very useful in applications. As an illustration of this fact, we study the relationship between 2-dimensional and chiral conformal AQFTs. Our skeletal models both for 2-dimensional and for chiral conformal AQFTs allow us to construct in Theorem 5.5 two adjunctions between the category of 2-dimensional conformal AQFTs and the category of chiral conformal AQFTs, whose right adjoints admit an interpretation as 'chiralization functors', i.e. they extract the ±-chiral components of a 2-dimensional conformal AQFT. This construction is a categorical formalization and also a generalization to the locally covariant setting of Rehren's chiral observables [Reh00]. Furthermore, the following structural result about the relationship between chiral and full AQFTs is proven: The category of chiral conformal AQFTs is a full coreflective subcategory of the category of 2-dimensional conformal AQFTs. In particular, this entails that the above adjunctions can be used to detect which 2-dimensional conformal AQFTs are chiral. The outline for the remainder of this paper is as follows: Section 2 provides a self-contained review of the general framework of AQFTs on orthogonal categories [BSW21] in which most of our statements and proofs are written. (Whenever possible, we avoid using operad theory.) In Section 3 we compute an explicit and very simple description of the localization of the category CLoc 2 at all Cauchy morphisms, which culminates in Theorem 3.9. Section 4 provides details for how to pass between the (equivalent) ordinary and skeletal descriptions of 2-dimensional conformal AQFTs. The passage from ordinary to skeletal is very simple, see Theorem 4.1. The reconstruction of the ordinary description from the skeletal one always exists but it is computationally much more involved. Under additional assumptions on the target category Alg, which are satisfied for * -algebras but not for C * -algebras, operadic left Kan extensions provide a concrete model for the reconstruction functor, see Theorem 4.3. In Section 5 we construct chiralization functors that extract the ±-chiral components of 2-dimensional conformal AQFTs and investigate their categorical properties, see Theorem 5.5. Finally, our chiralization functors are applied to a concrete example in Section 6, which illustrates how the chiralization of the Abelian current is related to the usual chiral currents. Appendix A proves a technical result that is used in Section 5.

Orthogonal categories and AQFTs
We shall briefly recall some relevant definitions and constructions in algebraic quantum field theory (AQFT). In order to state and prove the results of this paper, it will be crucial to work within the general framework of AQFTs on orthogonal categories [BSW21]. Such theories are most elegantly formulated via operads, however in order to make our results better accessible to a broader audience we shall provide a more elementary and self-contained description below. Our main constructions and results are insensitive to specific details of the target category in which the AQFTs take values. In particular, they hold true for * -algebras, locally convex * -algebras, bornological * -algebras, convenient * -algebras, C * -algebras and von Neumann algebras. In order to avoid excluding cases that might be of interest to some readers, we work with the following rather general choice of target category.
Definition 2.1. We fix once and for all an involutive symmetric monoidal category T and a (not necessarily full) subcategory Alg ⊆ * Alg As (T) of the category of associative and unital * -algebras in T, see e.g. [Jac12,BSW19]. We assume that the category Alg is complete, i.e. it admits all small limits.
Example 2.2. Let us briefly explain how our setting covers the standard choices of target categories. The category * Alg C = * Alg As (Vec C ) of associative and unital * -algebras over C is obtained by choosing the involutive symmetric monoidal category T = Vec C of complex vector spaces. Choosing instead the involutive symmetric monoidal category T = Ban C of Banach spaces, we obtain the complete category C * Alg C ⊆ * Alg As (Ban C ) of C * -algebras as a full subcategory of the category of Banach * -algebras. Furthermore, von Neumann algebras and normal unital * -homomorphism form a complete (but not full) subcategory W * Alg C ⊆ C * Alg C ⊆ * Alg As (Ban C ), see e.g. [Kor17]. ▽ The next definition formalizes a general concept of "spacetime category" in which certain pairs of morphisms f 1 : M 1 → M ′ ← M 2 : f 2 to a common target are distinguished. Physically, the distinguished pairs of morphisms should be interpreted as "causally independent subregions" in M ′ .
). An orthogonal category is a pair C = (C, ⊥) consisting of a small category C and a subset ⊥ ⊆ Mor C t × t Mor C of the set of pairs of morphisms to a common target, such that the following conditions hold true: We often denote elements ( We denote by OrthCat the 2-category whose objects are orthogonal categories, morphisms are orthogonal functors and 2-morphisms are natural transformations between orthogonal functors.
Example 2.4. The prime example Loc m = (Loc m , ⊥) of an orthogonal category is the usual category Loc m of oriented and time-oriented globally hyperbolic Lorentzian manifolds (of a fixed dimension m ≥ 2) with (f 1 : if and only if the images f 1 (M 1 ) and f 2 (M 2 ) are causally disjoint subsets of M ′ . See e.g. [BFV03,FV15] for more details. In the present work we will study various orthogonal categories associated with the one of connected oriented and time-oriented globally hyperbolic conformal Lorentzian 2-manifolds CLoc 2 . The latter was introduced in [Pin09] and used for conformal field theory studies in [CRV21]. We will give a precise definition of CLoc 2 in Section 3. ▽ The following definition formalizes the concept of AQFT on an orthogonal category C.
Definition 2.5 ([BSW21, Definition 3.5]). Let C be an orthogonal category. An AQFT on C is a functor A : C → Alg that satisfies the ⊥-commutativity property: For all (f 1 : M ′ denotes the (opposite) multiplication of the algebra A(M ′ ). We denote by AQFT(C) ⊆ Fun(C, Alg) the full subcategory of ⊥-commutative functors.
Remark 2.6. In the case of Loc m , ⊥-commutativity is also called Einstein causality. Note that Definition 2.5 does not explicitly mention the time-slice axiom [BFV03,FV15]. We shall show at the end of this section that the latter can be implemented through a localization of orthogonal categories. △ The assignment C → AQFT(C) of the AQFT categories can be promoted to a 2-functor where OrthCat is the 2-category introduced in Definition 2.3 and Cat denotes the 2-category of (not necessarily small) categories. This 2-functor assigns to an orthogonal functor F : C → D the pullback functor given by pre-composition with F . To a natural transformation ζ : F → G between orthogonal functors F, G : C → D it assigns the natural transformation (−)ζ : (−)F → (−)G obtained by whiskering. As every 2-functor, (2.2) preserves the equivalences in the respective 2-categories. The equivalences in Cat are the fully faithful and essentially surjective functors and the equivalences in OrthCat can be characterized as follows. Proof. "⇐": Since the underlying functor F : C → D is fully faithful and essentially surjective, it is an equivalence in the 2-category Cat, i.e. there exists a functor G : D → C and natural isomorphisms ε : F G → id D and ν : id C → GF . To obtain an equivalence in OrthCat, we have to prove that G : D = (D, ⊥ D ) → C = (C, ⊥ C ) is an orthogonal functor. Given any orthogonal pair (g 1 : , we obtain by using the natural isomorphism ε that F G(g i ) = ε −1 N ′ g i ε N i , for i = 1, 2, hence composition stability of ⊥ D entails that the pair F G(g 1 ) ⊥ D F G(g 2 ) is orthogonal. By definition of ⊥ C = F * (⊥ D ), it then follows that G(g 1 ) ⊥ C G(g 2 ) is orthogonal, which implies that G is an orthogonal functor. "⇒": By hypothesis, there exists an orthogonal functor G : D → C and natural isomorphisms ε : F G → id D and ν : id C → GF . This in particular implies that the underlying functor F : C → D is fully faithful and essentially surjective. It remains to check that, given f i : The conclusion follows by recalling that G is an orthogonal functor and that ⊥ C is composition stable.
In full generality, the pullback functor (2.3) does neither admit a left adjoint nor a right adjoint functor. The following result provides sufficient conditions for the existence of adjoint functors.  In the first and last step we have used that the AQFT categories are by Definition 2.5 full subcategories of the functor categories. The second step follows from the fact that Ran F is right adjoint to the pullback functor F * : Fun(D, Alg) → Fun(C, Alg).
Remark 2.9. The hypotheses of item a) are satisfied for the category * Alg C = * Alg As (Vec C ) of associative and unital * -algebras over C, but they are not satisfied for the category C * Alg C ⊆ * Alg(Ban C ) of C * -algebras. It is important to emphasize that the left adjoints from item a) are not necessary for proving any of the structural results of this paper. We will only use them in Section 4 to determine a concrete model for a quasi-inverse of an equivalence between AQFT categories. △ We conclude this section with a discussion of the time-slice axiom, which in our general setup from Definition 2.5 takes the following form.
-commutative functor for the pushforward orthogonality relation because the latter is generated by ) and a natural isomorphism L * (A) ∼ = B, hence (2.7) is essentially surjective.

Localization of CLoc 2 at Cauchy morphisms
The goal of this section is to develop an explicit and simple model for the localization of the orthogonal category CLoc 2 of oriented and time-oriented globally hyperbolic conformal Lorentzian 2-manifolds at all Cauchy morphisms. Via Proposition 2.12, this will provide us with a very efficient description of 2-dimensional conformal AQFTs satisfying the time-slice axiom, which we will exploit later in the following sections.
Let us start by recalling the category CLoc 2 and its orthogonal structure, see also [Pin09,CRV21] for earlier appearances of this category.
Definition 3.1. The category CLoc 2 is defined as follows: Its objects are all oriented and timeoriented Lorentzian 2-manifolds M that are globally hyperbolic and connected. 2 A morphism f : M → M ′ is an orientation and time-orientation preserving embedding with causally convex image f (M ) ⊆ M ′ that preserves the conformal structure determined by the metrics, i.e. f * g ′ = Ω 2 g for some conformal factor Ω 2 ∈ C ∞ (M, R >0 ). The orthogonal category CLoc 2 := (CLoc 2 , ⊥) is then defined as follows: A pair of morphisms is orthogonal (f 1 : We denote by W ⊆ Mor CLoc 2 the set of all Cauchy morphisms.
Remark 3.2. We decided to restrict ourselves to connected manifolds in order to simplify the presentation of this paper. Our results and proofs do generalize in a fairly obvious way to nonconnected manifolds by treating each connected component separately. △ Example 3.3. There are two distinguished objects in CLoc 2 that will play a prominent role in our construction. First, we have the 2-dimensional Minkowski spacetime, which we shall denote by M ∈ CLoc 2 . Explicitly, the underlying manifold of M is given by R 2 , which we describe by the two light-cone coordinates x ± = t ± x. The metric, orientation and time-orientation then read as Note that every causally convex, connected and open subset U ⊆ M defines an object U ∈ CLoc 2 when endowed with the restricted metric, orientation and time-orientation. Furthermore, the inclusion map ι M U : U → M is manifestly a CLoc 2 -morphism.
The second distinguished object of interest to us is the 2-dimensional flat cylinder, which we denote by M /Z ∈ CLoc 2 . Explicitly, the cylinder may be obtained as a quotient of M by the Z-action Z × M → M , (n, x ± ) → x ± ± n, together with the induced metric, orientation and time-orientation. As before, every causally convex, connected and open subset V ⊆ M /Z defines an object V ∈ CLoc 2 when endowed with the restricted metric, orientation and time-orientation and the inclusion map ι Before we can address the problem of localizing CLoc 2 at all Cauchy morphisms, we have to develop a better understanding of the objects and morphisms in this category. The first step of our approach consists of using known conformal embedding theorems for globally hyperbolic Lorentzian 2-manifolds in order to obtain an equivalent category C 2 ≃ CLoc 2 that is easier to work with. For this we recall that, as a consequence of global hyperbolicity and connectedness, there exist two distinct types of objects M ∈ CLoc 2 : The Cauchy surfaces of M are either diffeomorphic to the line R or to the circle T = R/Z. Note that the Minkowski spacetime M ∈ CLoc 2 is of the first type and the flat cylinder M /Z ∈ CLoc 2 is of the second type. The following embedding results are (essentially, see the proof below) proven in [  It remains to show that the image V ⊆ M /Z of the embedding M → M /Z is causally convex, which we will prove by contraposition. Suppose that V ⊆ M /Z is not causally convex. Then there exists a causal curve γ : [0, 1] → M /Z from γ(0) ∈ V to γ(1) ∈ V that exits and re-enters V . Extending γ to an inextensible causal curveγ : R → M /Z, it meets precisely once the space-like Cauchy surface Σ ′ in the image V ⊆ M /Z that we have constructed above. Co-restrictingγ to V ⊆ M /Z we obtain an immersionγ| :γ −1 (V ) → V . The domainγ −1 (V ) ⊆ R has at least two connected components, because by hypothesis the causal curve exits and re-enters V . By restricting to the connected components,γ| defines at least two inextensible causal curves in V , of which however only one meets the space-like Cauchy surface Σ ′ . Therefore, inverting the conformal embedding M → V ⊆ M /Z onto V , one finds an inextensible causal curve in M that does not meet Σ. This contradicts the hypothesis that Σ is a space-like Cauchy surface of M .
Let us endow C 2 ⊆ CLoc 2 with the pullback along the inclusion of the orthogonality relation ⊥ on CLoc 2 and denote the resulting orthogonal category by C 2 := (C 2 , ⊥). Combining Lemma 2.7 and the equivalence of categories from Corollary 3.5, it follows that the inclusion defines an orthogonal equivalence C 2 ∼ −→ CLoc 2 . Furthermore, a morphism in C 2 is Cauchy if and only if its image in CLoc 2 is such. This entails that passing to the orthogonal localizations at all Cauchy morphisms both in C 2 and in CLoc 2 defines an equivalence of orthogonal categories . The next goal is to find an explicit model for the orthogonal localization C 2 [W −1 ] at the subset W ⊆ Mor C 2 of all Cauchy morphisms in C 2 . Our strategy is to construct, similarly to [BDS18], a reflective localization by using Cauchy developments in the ambient spacetimes M and M /Z. Let us recall that, given any object M ∈ CLoc 2 , the Cauchy development of a subset S ⊆ M is the subset D(S) ⊆ M of all points p ∈ M such that every inextensible causal curve through p meets S.
Next, let us consider the case of a The conformal embedding f : V → V ′ lifts to a conformal immersionf :Ṽ →Ṽ ′ between the universal covers. One easily checks thatf is of the formf (x + , x − ) = (f + (x + ),f − (x − )) withf ± : pr ± (Ṽ ) = R → pr ± (Ṽ ′ ) = R two orientation preserving embeddings satisfying the Z-equivariance conditioñ f ± (y + 1) =f ± (y) + 1, for all y ∈ R. Passing to the quotients then defines the desired CLoc 2morphism D(f ) :=f The resulting functor D : C 2 → C D 2 is left adjoint to the inclusion functor i : C D 2 → C 2 . The adjunction unit η : id C 2 → i D is given by the components η U := ι Theorem 3.9. The various orthogonal categories constructed in this section are related by the following diagram of orthogonal functors that commutes up to the displayed natural isomorphisms. In this diagram equivalences of orthogonal categories are labeled by ∼. The symbol ǫ denotes the counit of the reflective localization D ⊣ i from Proposition 3.7 and L : CLoc 2 is the orthogonal localization functor that is obtained by choosing any quasi-inverse of the equivalence C 2 ∼ −→ CLoc 2 .
We conclude this section by describing the orthogonal category C D,skl 2 more explicitly. By definition, this category has only two objects, the Minkowski spacetime M and the flat cylinder M /Z. Using similar arguments as in the proof of Proposition 3.7, we can also describe the corresponding Hom-sets. Denote by Emb + (R) the set of orientation preserving embeddings of R into itself. For the endomorphisms of Minkowski spacetime, we find a bijection (3.2a) Explicitly, the CLoc 2 -morphism associated with a pair (f + , f − ) ∈ Emb + (R) 2 of orientation preserving embeddings of R into itself reads as f : Furthermore, denote by Diff + (T) the set of orientation preserving diffeomorphisms of T = R/Z. For the endomorphisms of the flat cylinder, we find the bijection given by associating to a pair (g + , g − ) ∈ Diff + (T) 2 of orientation preserving diffeomorphisms of T = R/Z the CLoc 2 -automorphism g : Here the Z-action on Emb +,≤1 (R) 2 is given by translation To characterize the orthogonality relation on C D,skl 2 , let us first note that the causal future/past of a (possibly unbounded) double cone subset I + × I − := (a + , b + ) × (a − , b − ) ⊆ M is given by is given explicitly as follows: (iii) (g + , g − ) : M /Z → M /Z is not orthogonal to any morphism.

Skeletal model and reconstruction
As a consequence of Theorem 3.9 and Proposition 2.12, the two-object orthogonal category C D,skl 2 captures the theory of 2-dimensional conformal AQFTs satisfying the time-slice axiom, in the sense that we have an equivalence of categories AQFT(CLoc 2 ) W ≃ AQFT(C D,skl 2 ). (See Theorem 4.1 below for the precise statement.) The latter perspective is very efficient: By Definition 2.5, a theory A ∈ AQFT(C D,skl 2 ) simply consists of two algebras, one for the Minkowski spacetime A(M) ∈ Alg and one for the flat cylinder A( M /Z) ∈ Alg, together with a ⊥-commutative action of the morphisms in C D,skl 2 . We refer to this description of 2-dimensional conformal AQFTs satisfying time-slice as the skeletal model.
The aim of this section is to spell out in more detail how to pass between the ordinary description and the skeletal one. Composing the horizontal orthogonal functors in the diagram (3.1) of Theorem 3.9, we obtain the orthogonal full subcategory inclusion to the full subcategory of 2-dimensional conformal AQFTs satisfying time-slice is an equivalence of categories.
Proof. Observe that the orthogonal functor j defined in (4.1) is the composition of the horizontal orthogonal functors in (3.1). Applying the AQFT 2-functor (2.2) to the diagram (3.1), we obtain a diagram of functors that commutes up to the displayed natural isomorphisms. (As before, we label equivalences by ∼.) Using now Proposition 2.12 for the orthogonal localization functors L and D, we obtain the diagram in which each functor is an equivalence. The composition of the horizontal functors coincides with the restricted pullback functor j * in (4.2), hence we have shown that the latter is an equivalence. Spelling out the reconstruction of the ordinary description of a 2-dimensional conformal AQFT satisfying time-slice from a skeletal model is more involved because it requires finding a quasiinverse for the equivalence in (4.2). It is important to emphasize that every equivalence of categories does admit a quasi-inverse, hence the question here is not about the existence of a reconstruction functor but rather about finding a concrete model. We shall now solve this problem under the additional hypothesis that the target category Alg = * Alg As (T) is the category of associative and unital * -algebras in a cocomplete involutive closed symmetric monoidal category T. (Recall from Remark 2.9 that this is the case for the category * Alg C = * Alg As (Vec C ) of associative and unital * -algebras over C, but it is not the case for the category of C * -algebras.) Using the left adjoint functors from item a) of Proposition 2.8 associated with the horizontal orthogonal equivalences in (3.1), which we collectively denote by !, we define the composite reconstruction functor rec : AQFT(C D,skl where D * denotes the pullback along the orthogonal localization functor D : Theorem 4.3. Suppose that Alg = * Alg As (T) is the category of associative and unital * -algebras in a cocomplete involutive closed symmetric monoidal category T. Then the functor (4.5) takes values in the full subcategory AQFT(CLoc 2 ) W ⊆ AQFT(CLoc 2 ) of 2-dimensional conformal AQFTs satisfying time-slice and it defines a quasi-inverse of (4.2).
Proof. Recall that the pullback functor j * in (4.2) is given by the composite of the first row in the diagram (4.4) and that by construction the functor (4.5) is the composite of the left adjoints of the functors displayed in the first row of (4.3). In particular, the first functor in (4.5) is an equivalence, D * : AQFT(C D 2 ) → AQFT(C 2 ) W ⊆ AQFT(C 2 ) provides an equivalence onto the full subcategory of W -constant AQFTs by Proposition 2.12 and the last functor in (4.5), which we denote in this proof by k ! : AQFT(C 2 ) → AQFT(CLoc 2 ), is part of an adjoint equivalence, whose right adjoint functor is displayed on the top right of (4.3). Since the latter functor preserves W -constancy, it follows that this adjoint equivalence restricts to the full subcategories AQFT(C 2 ) W ⊆ AQFT(C 2 ) and AQFT(CLoc 2 ) W ⊆ AQFT(CLoc 2 ) of W -constant AQFTs if k ! preserves W -constancy too, i.e. k ! sends W -constant AQFTs on C 2 to W -constant AQFTs on CLoc 2 .
To prove the latter statement, let A ∈ AQFT(C 2 ) W be any W -constant AQFT on C 2 . By Proposition 2.12, such A ∼ = D * (B) is isomorphic to the pullback along the localization functor D of some B ∈ AQFT(C D 2 ). We then compute where in the second step we have used the right triangle in (4.3) (the horizontal arrow is k * according to the notation used in this proof) and the third step follows from the fact that k ! is left adjoint to the equivalence k * . Applying again Proposition 2.12, but now for the orthogonal localization functor L, proves that k ! (A) ∼ = L * (B) ∈ AQFT(CLoc 2 ) W is W -constant.
Remark 4.4. While the pullback functor D * in (4.5) is easy to compute, the two left adjoints ! are more involved. Using standard techniques from operad theory, see e.g. [BSW21, Proposition 2.12] and also [BSW19, Section 6] for the * -operadic case, it is possible to provide point-wise colimit formulas for both instances of !. In particular, this means that, given any skeletal model ), reconstructing its ordinary description rec(A) ∈ AQFT(CLoc 2 ) W requires computing, for each object M ∈ CLoc 2 , a double colimit rec(A)(M ) = ! D * !(A)(M ) ∈ T in the target category T. Since these explicit colimit formulas are not very instructive, we shall not spell them out in detail. △

Chiralization adjunctions
In this section we study the relationship between 2-dimensional conformal AQFTs that satisfy the time-slice axiom and chiral conformal AQFTs. Describing both types of theories via their skeletal models, we will construct adjunctions that allow us to assign to each chiral conformal AQFT a 2-dimensional conformal AQFT satisfying time-slice, and vice versa assign to each 2-dimensional conformal AQFT satisfying time-slice its two chiral components. Let us first introduce the relevant orthogonal category that, via Definition 2.5, defines the category of chiral conformal AQFTs.
The orthogonality relation on Man skl 1 then reads explicitly as follows: (iii) g : T → T is not orthogonal to any morphism.
Comparing this to our explicit description of the orthogonal category C D,skl 2 (see the end of Section 3), we observe that there exist two evident orthogonal functors π ± : C D,skl 2 −→ Man skl 1 (5.3a) that act on objects as π ± (M) = R , π ± ( M /Z) = T , (5.3b) and on morphisms by projecting onto the ±-component, i.e. and on morphisms we have that Hence, we clearly see that π ± * (B) is only sensitive to one of the light-cone coordinates x ± , which is the characteristic feature of a chiral theory.
Suppose for the moment that our target category Alg satisfies the hypotheses of item a) in Proposition 2.8. Then there exist left adjoint functors π ± ! : AQFT(C D,skl 2 ) −→ AQFT(Man skl 1 ) (5.6) that allow us to map from 2-dimensional conformal AQFTs satisfying time-slice to chiral conformal AQFTs. It is tempting to think of π ± ! as a "chiralization functor" that extracts the ±-chiral component of a 2-dimensional conformal AQFT. However, this functor is not suitable for this task because, in many important cases, it yields trivial theories. Let us substantiate this claim.
Example 5.3. Suppose that the hypotheses of item a) in Proposition 2.8 are satisfied. (For instance, we can take Alg = * Alg C , the category of associative and unital * -algebras over C.) Then the operadic left Kan extension π ± ! (A) ∈ AQFT(Man skl 1 ) of a theory A ∈ AQFT(C D,skl 2 ) can be worked out by using the explicit model from [BSW21, Proposition 2.12]. Evaluating π ± ! (A) on the object R ∈ Man skl 1 , one then finds that is the quotient of the Minkowski spacetime algebra A(M) ∈ Alg by a two-sided ideal I ∓ ⊆ A(M).
The ideal I − is generated by the elements A(id, k)(a) − a, for all k ∈ Emb + (R) and a ∈ A(M), and the ideal I + is generated by the elements A(k, id)(a) − a, for all k ∈ Emb + (R) and a ∈ A(M).
In words, this means that π ± ! (A)(R) is the algebra of coinvariants of A(M) associated with the action of the embedding monoid Emb + (R) of the opposite chirality.
We are now in the position to explain why π ± ! does not provide a sensible chiralization functor. Recall that many important examples in AQFT, e.g. the free theories constructed via CCR (or CAR) quantization of non-degenerate Poisson (or inner product) vector spaces, are described by simple algebras. So let us suppose that the theory A assigns a simple algebra A(M) ∈ Alg to the Minkowski spacetime M. Then the quotient algebra (5.7) that is assigned to the line R is either the trivial algebra 0 or A(M), depending on whether the two-sided ideal I ∓ ⊆ A(M) is all of A(M) or 0. The latter case I ∓ = 0 arises if and only if the theory A is insensitive to the light-cone coordinate x ∓ of Minkowski spacetime M, i.e. A(id, k) = id for all k ∈ Emb + (R) in the case of − and A(k, id) = id for all k ∈ Emb + (R) in the case of +, which is only true in the very restrictive case where A is chiral. It follows that π ± ! (A)(R) ∼ = 0 is the trivial algebra for many important examples of non-chiral 2-dimensional conformal AQFTs, including in particular the free scalar field (see e.g. [CRV21]) or the Abelian current from Section 6, which explains our claim that π ± ! do not admit the interpretation of "chiralization functors". ▽ We will now show that the hypotheses of item b) in Proposition 2.8 are satisfied in the present case (see Theorem 5.5 below), hence we obtain right adjoint functors π ± * : AQFT(C D,skl 2 ) −→ AQFT(Man skl 1 ) . (5.8) We will argue in Remark 5.7, and further illustrate by a concrete example in Section 6, that the latter define physically sensible chiralization functors. Before we can apply item b) of Proposition 2.8, we have to develop an explicit model for Ran π ± . Let us consider first the simpler case N = T. The under category then reads as T ↓ π + ≃ Obj : g ∈ Diff + (T) Mor : Diff + (T) 2 ∋ (g + , g − ) : g → g + g (5.11) and the forgetful functor assigns g → M /Z and ((g + , g − ) : Introducing the category BDiff + (T) consisting of a single object * with morphisms Diff + (T), one easily checks that the functor is initial. (The relevant argument is completely analogous to the "simple case" in Appendix A.) This implies that (5.10) for N = T is isomorphic to the limit Ran π + (A)(T) ∼ = lim BDiff + (T) Rephrasing this result in a more concrete language, we obtain that is the algebra of invariants of A( M /Z) associated with the action of the diffeomorphism group Diff + (T) of the opposite chirality.
Let us consider now the case N = R, in which the under category is richer .
(5.15) Introducing the category BEmb + (R) consisting of a single object * with morphisms Emb + (R), one checks that the functor is initial. (This check is more involved than in the previous case N = T. The relevant details can be found in Appendix A.) This implies that (5.10) for N = R is isomorphic to the limit It remains to describe the action of the functor Ran π + (A) : Man skl 1 → Alg on morphisms. For the case of (h : R → R) ∈ Emb + (R), one finds For the case of ([h] : R → T) ∈ Emb +,≤1 (R) Z, we obtain where k ∈ Emb +,≤1 (R) is chosen arbitrarily. Using the zig-zags constructed in Appendix A, one immediately checks that the morphism Ran π + (A)( This completes our description of the categorical right Kan extension Ran π + for π + . The case Ran π − for π − is completely analogous by swapping the two chiralities. Proof. Using the model for Ran π ± from Construction 5.4, one easily checks that Ran π ± (A) : Man skl 1 → Alg is a ⊥-commutative functor for all ⊥-commutative functors A : C D,skl 2 → Alg. Hence, item b) of Proposition 2.8 proves the first statement.
Remark 5.6. The counit ǫ : π ± * π ± * → id of the adjunction π ± * ⊣ π ± * admits an explicit description too. Let us spell out the details for π + and note that the case of π − is completely analogous by swapping the two chiralities. For all A ∈ AQFT C D,skl 2 ), we find using again (5.5), (5.14), (5.18) and (5.19) that and that where in the second steps we use explicitly that these morphisms act on invariants. The component ǫ A : π + * π + * (A) → A of the counit is then given by including the subalgebras of invariants. Note that, in contrast to the unit, the components of the counit are in general not isomorphisms. A necessary and sufficient condition for ǫ A to be an isomorphism is that A(M) inv − = A(M) and A( M /Z) inv − = A( M /Z). This is the case if and only if A is insensitive to the light-cone coordinate x − , i.e. A is +-chiral. In other words, the counits of the adjunctions (5.20) allow us to detect whether or not a 2-dimensional conformal AQFT A ∈ AQFT C D,skl 2 ) is chiral: Indeed, A is ±-chiral if and only if the corresponding component of the counit ǫ A : π ± * π ± * (A) → A is an isomorphism. △ Remark 5.7. The right adjoint functors π ± * of the adjunctions in Theorem 5.5 should be interpreted as chiralization functors that extract the ±-chiral components of a 2-dimensional conformal AQFT A ∈ AQFT C D,skl 2 ) satisfying the time-slice axiom. Our construction provides an elegant categorical formalization, and also a generalization to the context of locally covariant conformal AQFTs, of an earlier proposal by Rehren [Reh00] who has defined the chiral components of a 2-dimensional local conformal net on the Minkowski spacetime M. (In our terminology, this is an AQFT on the category of all double cone subsets I + × I − ⊆ M with orthogonality relation given by causal disjointness.) The (maximal) chiral observable algebras are defined in [Reh00, Definition 2.1] by first extending the 2-dimensional theory to a covering manifold of M, which is isomorphic to the cylinder [BGL93], and then taking invariants of the (vacuum preserving) Möbius subgroup Möb ⊂ Diff + (T) of the diffeomorphism group of the opposite chirality. Under additional assumptions on the 2-dimensional theory, the chiral observable algebras also admit a more geometrical description by taking intersections of 2-dimensional double cone algebras [Reh00, Corollary 2.7]. △ 6 Example: The Abelian current We illustrate our chiralization construction from Theorem 5.5 by applying it to the 2-dimensional conformal AQFT that describes the Abelian current. In particular, we will show that the resulting chiral components are related to the usual chiral currents. In this section we choose Alg = * Alg C to be the category of associative and unital * -algebras over C.
where in the third step we have used the identity Chiralization on the Minkowski spacetime: Working in light-cone coordinates x ± on the Minkowski spacetime M = M, one finds that the (anti-)self-dual 1-forms are given by Ω 1,± c (M) = C ∞ c (M) dx ± . Using fiber integrations along the projection maps π ± : M → R, we obtain a linear isomorphism it is a compactly supported function of the light-cone coordinate x + . Similarly, the fiber integration of [ϕ dx + ] ∈ Ω 1,+ c (M)/d + M C ∞ c (M) is a compactly supported function of x − . This means that one should associate the first summands in (6.10) with + and the second summands with −. This is clarified further by studying the action of the endomorphisms Hom CLoc 2 (M, M) ∼ = Emb + (R) 2 (see also (3.2)) on the vector space L ′ (M) that is induced via the isomorphism (6.10) from the action on L(M). For (f + , f − ) ∈ Emb + (R) 2 , one finds from the fiber-wise diffeomorphism invariance of fiber integrations that where f ± * denote the pushforwards of compactly supported functions. Hence, the first summand in L ′ (M) transforms under f + and the second summand transforms under f − .
Let us also describe the Poisson structure (6.6) from the isomorphic perspective of L ′ (M). First, let us note that the causal propagator for P M = 2 dx − ∧ dx + ∂ − ∂ + acts on a compactly supported 2-form ω = ρ dx − ∧ dx + ∈ Ω 2 c (M) as where sgn denotes the sign function defined by sgn(x) = 1 for x > 0, sgn(x) = 0 for x = 0, and sgn(x) = −1 for x < 0. Inserting this into (6.6), one directly checks that (6.13) Hence, the induced Poisson structure on L ′ (M) reads as (6.14) With these preparations, we can now prove the main result of this paragraph.
Proposition 6.1. The chiral components π ± * (A)(R) ∈ * Alg C of the Abelian current are , for all k ∈ Emb + (R). Recall further that invariants are categorical limits, which are created through the forgetful functor at the level of the underlying vector spaces. Passing via the isomorphism (6.10) to the simplified description L ′ (M), we find for the underlying vector spaces that where in the last step we have used that due to (6.11) the morphisms A(id, k) only act non-trivially on the second summand of L ′ (M) and that the tensor product ⊗ C is exact, hence it commutes with forming invariants. One easily checks that In more detail, any element a ∈ Sym C C ∞ c (R) can be represented as a finite sum a = N n=0 a n , where a n ∈ C ∞ c (R n , C) is a compactly supported complex-valued function on the product manifold R n . (Note that a 0 is a function on the point R 0 = pt, which is the same datum as a constant a 0 ∈ C.) Such a is invariant under the diagonal action of translations if and only if all a n are invariant. Due to the compact supports, this is the case if and only if a n = 0 for all n ≥ 1, which proves the claim that Sym C C ∞ c (R) inv − ∼ = C.
Summing up, we find that, from the isomorphic perspective L ′ (M) = C ∞ c (R) ⊕ C ∞ c (R), the algebra A(M) inv − is generated by the subspace C ∞ c (R) ⊕ 0 ⊆ L ′ (M). Under the isomorphism (6.10), this gives the subspace L + (M) ⊆ L(M) defined in (6.16), which completes the proof.
Remark 6.2. The chiral components π ± * (A)(R) ∈ * Alg C from Proposition 6.1 coincide with the usual chiral currents on R. Indeed, using again the isomorphism (6.10), we find where by (6.14) the Poisson structure reads as Chiralization on the flat cylinder: The case of the flat cylinder M = M /Z is broadly similar to the Minkowski spacetime. For completeness, we shall spell out the relevant details. We work again in light-cone coordinates x ± , which in the cylinder case are subject to the identification (x + + 1, x − − 1) ∼ (x + , x − ) arising from the quotient by the Z-action. The (anti-)self-dual 1forms are given by Ω 1,± c ( M /Z) = C ∞ c ( M /Z) dx ± . The Minkowski projection maps π ± : M → R are Z-equivariant, hence they define fiber bundles π ± : M /Z → T = R/Z over the circle whose fibers are the ±-light rays in M /Z. (In particular, the fibers are diffeomorphic to R.) Using again fiber integrations, we obtain a linear isomorphism (6.20) Injectivity follows from the Poincaré lemma for compact vertical supports, see [BT82, Proposition 6.16]. Let us spell out the proof that the pr ± * are indeed surjective maps. It is sufficient to consider the case pr + * since the case pr − * follows by a similar argument. Let ϕ ∈ C ∞ (T) be any smooth function, which we regard as a Z-invariant function ϕ ∈ C ∞ (R) Z on the real line. Take any compactly supported function ρ ∈ C ∞ c (R) such that R ρ(z) dz = 1 and define the anti-self-dual 1-form α := ρ(x − + x + ) ϕ(x + ) dx − ∈ Ω 1,− (M) on Minkowski spacetime. Note that α is invariant under the Z-action (x + , x − ) → (x + + n, x − − n) and that its support is time-like compact. Hence, it descends to a compactly supported anti-self-dual 1-form α ∈ Ω 1,− c ( M /Z) on the cylinder. Applying fiber integration we find pr + * (α)(x + ) = R ρ(x − + x + ) ϕ(x + ) dx − = ϕ(x + ) R ρ(z) dz = ϕ(x + ) , (6.21) where in the second step we have changed the integration variable according to z := x − + x + . This proves surjectivity of pr + * . The induced action of the endomorphisms Hom CLoc 2 ( M /Z, M /Z) ∼ = Diff + (T) 2 on the isomorphic vector space L ′ ( M /Z) reads as for all (g + , g − ) ∈ Diff + (T) 2 .
We will now show that, from the isomorphic perspective L ′ ( M /Z), the Poisson structure (6.6) is given by To prove this claim, we use a convenient description of the causal propagator GM /Z on the flat cylinder that is known as the 'method of images', see e.g. [CRV21, Appendix A] for more details. Any compactly supported 2-form ω ∈ Ω 2 c ( M /Z) on the cylinder can be regarded as a time-like compactly supported and Z-invariant 2-form ω ∈ Ω 2 tc (M) Z on the Minkowski spacetime. Due to the support properties of its integral kernel, the application G M (ω) of the Minkowski causal propagator (6.12) on such ω is well-defined and one directly checks that the result is Z-invariant, hence it defines a function on the cylinder which coincides with GM /Z (ω) ∈ C ∞ ( M /Z). The proof that (6.6) induces (6.23) is then analogous to the case of the Minkowski spacetime.
In order to state and prove the main result of this paragraph, we shall need one more ingredient. Observe that there exists an injective linear map Proof. It is sufficient to prove the result for π + * (A)(T) because π − * (A)(T) follows by the same argument upon swapping the two chiralities. Arguing in complete analogy to the proof of Proposition 6.1, we find for the underlying vector spaces that (6.28) Denoting by R ⊆ C ∞ (T) the subspace of the constant functions, we have an inclusion Sym C R ⊆ Sym C C ∞ (T) inv − because each constant function is diffeomorphism invariant. Let us show that Sym C R = Sym C C ∞ (T) inv − are equal. Any invariant element a ∈ Sym C C ∞ (T) inv − can be represented as a finite sum a = N n=0 a n , where a n ∈ C ∞ (T n , C) is a complex-valued function on the n-torus T n that is invariant under the diagonal Diff + (T)-action. (Note that a 0 is a function on the point T 0 = pt, which is the same datum as a constant a 0 ∈ C.) We now claim that each a n is constant, which can be proven locally by restricting to a sufficiently small open neighborhood of an arbitrary point (p 1 , . . . , p n ) ∈ T n . For this we consider the open subset U = n i=1 (p i − 1 4 , p i + 1 4 ) ⊆ T n and pick any diagonal diffeomorphism R n ∼ = −→ U . By restriction and pullback, we obtain a functionã n ∈ C ∞ (R n , C) that is invariant under the diagonal action of Diff + (R). In particular,ã n (λ x 1 , . . . , λx n ) =ã n (x 1 , . . . , x n ) for all dilations λ ∈ R ≥0 , which implies thatã n is constant. This implies that a n is locally constant around any point, hence a n is constant.
Summing up, we find that, from the isomorphic perspective L ′ ( M /Z) = C ∞ (T) ⊕ C ∞ (T), the algebra A( M /Z) inv − is generated by the subspace C ∞ (T) ⊕ R ⊆ L ′ ( M /Z). Under the isomorphism (6.20), this gives the subspace L + ( M /Z) ⊆ L( M /Z) defined in (6.27), which completes the proof.
Remark 6.4. The chiral components π ± * (A)(T) ∈ * Alg C from Proposition 6.3 coincide with a tensor product of the usual chiral currents on T and a commutative algebra. Indeed, using again the isomorphism (6.20) and the observation that the Poisson structure (6.23) acts trivially on constant functions, we find π ± * (A)(T) ∼ = CCR C ∞ (T), τ T ⊗ C Sym C R , (6.29) where τ T (ϕ, ψ) = − 1 2 T ϕ d T ψ , (6.30) for all ϕ, ψ ∈ C ∞ (T). Recalling that the vector space R ∼ = H 1,± c ( M /Z) arises as the (anti-)selfdual compactly supported cohomology (6.25), the algebra Sym C R admits an interpretation as topological observables associated with the opposite chirality. Such topological observables are in particular diffeomorphism invariant, hence they survive our chiralization construction that is implemented by taking diffeomorphism invariants. △