The linear CS/WZW bulk/boundary system in AQFT

This paper constructs in the framework of algebraic quantum field theory (AQFT) the linear Chern-Simons/Wess-Zumino-Witten system on a class of $3$-manifolds $M$ whose boundary $\partial M$ is endowed with a Lorentzian metric. It is proven that this AQFT is equivalent to a dimensionally reduced AQFT on a $2$-dimensional manifold $B$, whose restriction to the $1$-dimensional boundary $\partial B$ is weakly equivalent to a chiral free boson.


Introduction and summary
Quantum field theories (QFTs) that are defined on manifolds with boundary are known to exhibit interesting phenomena that arise from a rich interplay between bulk and boundary degrees of freedom.A prime example is given by the Chern-Simons/Wess-Zumino-Witten (in short, CS/WZW) bulk/boundary system that describes a topological QFT in the bulk, a chiral conformal QFT on the boundary and their non-trivial interplay, leading to interesting phenomena such as holography.This was observed first in [Wit89] and subsequently studied in more detail by many authors, see e.g.[EMSS89; Gaw99; FFFS02; MSW19; P ŠV21; GRW22; CMW23].One of the main mechanisms causing an interplay between bulk and boundary degrees of freedom is gauge invariance of the combined bulk/boundary system.This leads to various consistency conditions between bulk and boundary fields that can be analyzed, for instance, in the BV/BFV formalism of Cattaneo, Mnev and Reshetikhin [CMR14;CMR18].
The aim of this paper is to study in detail a simple and explicit example of such gauge-theoretic bulk/boundary system from the point of view of algebraic QFT (AQFT) [HK64;BFV03], or more precisely its homotopical refinement [BSW19; BS19] that is better suited to deal with gauge theories.The model we study is given by (a Lorentz geometric variant of) the linear CS/WZW bulk/boundary system, which describes linear Chern-Simons theory in the bulk of a 3-manifold M and a chiral free boson on its boundary ∂M , together with their interplay.More precisely, we study Chern-Simons theory with linear structure group G = R on an oriented 3-manifold M , whose boundary ∂M is endowed with (the conformal class of) a Lorentzian metric and a time-orientation, in the presence of a Lorentzian version of the chiral WZW boundary condition on ∂M .A holomorphic variant of this example, where ∂M is endowed with a complex structure, was studied recently in the context of factorization algebras (à la Costello-Gwilliam [CG17; CG21]) by Gwilliam, Rabinovich and Williams [GRW22].
The physics of the CS/WZW bulk/boundary system is already well-understood, and by restricting to the linear structure group G = R we only study a particularly simple instance of it.Of interest in this work, though, are the mathematical techniques we use to construct and analyze the model as an AQFT.They relate to multiple recent developments in AQFT and differ considerably from how [GRW22] construct the linear CS/WZW system as a factorization algebra.Some of the key features are: (1) While the factorization algebra for the holomorphic CS/WZW system is constructed in [GRW22] by linear BV quantization of a shifted Poisson structure (the antibracket), our construction of an AQFT for the Lorentzian variant of this system uses canonical commutation relation (CCR) quantization of a related unshifted Poisson structure.We will determine the latter by using the novel concept of Green's homotopies from [BMS22a], which are homotopical generalizations of Green's operators, in combination with techniques from 2-dimensional chiral conformal Lorentzian geometry [Gra23].In particular, we provide new examples of Green's homotopies for gauge-theoretic bulk/boundary systems, which contributes novel types of examples to the literature on Green's operators in the presence of Lorentzian boundaries, see e.g.[DDF19; BDS18; DDL20].
(2) Our AQFT for the CS/WZW bulk/boundary system is defined on a suitable category of open subsets of the 3-manifold M and it satisfies a homotopical version of local constancy, or the time-slice axiom.Using recent strictification theorems for the homotopy time-slice axiom [BCS23], we will prove that this AQFT is equivalent to an AQFT that is defined on a 2-dimensional manifold B, playing a similar role to a Cauchy surface.This dimensional reduction from 3 to 2 dimensions is crucial to identify the degrees of freedom of the CS/WZW system near the boundary ∂B with a chiral free boson, which in the context of AQFT is described on 1-manifolds (i.e.light rays).
The outline of the remainder of this paper is as follows: In Section 2, we review the linear Chern-Simons complex on an oriented 3-manifold M with boundary ∂M = ∅ from [GRW22] and introduce our Lorentz geometric variant of the chiral WZW boundary condition, which we call the (anti-)self-dual boundary condition L ± , see (2.18).In Section 3, we define a suitable concept of chiral Green's homotopies for the Chern-Simons complex and prove their existence under suitable niceness conditions, see Proposition 3.28.The main tool we use to prove the existence of Green's homotopies is an extension of the ±-chiral flow Θ ± [Gra23] on the boundary Lorentzian 2-manifold ∂M to a proper R-action Θ ± : R × M → M on the bulk manifold.One may interpret this as an arbitrary extension of the concepts of ±-chiral null curves from ∂M into the bulk M .Equipped with these Green's homotopies, we construct in Section 4 an AQFT A ± for the CS/WZW bulk/boundary system that assigns differential graded * -algebras (in short, dg-algebras) to suitable open subsets U ⊆ M .We also prove that this AQFT satisfies a chiral variant of the Einstein causality axiom and of the homotopy time-slice axiom, see Proposition 4.6 and Corollary 4.7.
In Section 5, we use the strictification theorems for the homotopy time-slice axiom from [BCS23] in order to prove that our 3-dimensional AQFT A ± for the CS/WZW bulk/boundary system on M is equivalent to a 2-dimensional AQFT B ± that is defined on the quotient space B ± := M/ ∼ ± associated with the proper R-action Θ ± : R×M → M , see Corollary 5.5.(Choosing a smooth section of the associated principal R-bundle π ± : M → B ± allows one to think of B ± as a kind of "Cauchy surface" in M .)Using this phenomenon of dimensional reduction, we also prove that, under suitable topological assumptions on M , the AQFT A ± for the CS/WZW bulk/boundary system is insensitive to the choice of extension of the ±-chiral flow Θ ± on ∂M to a proper R-action Θ ± on M , see Corollary 5.9.Hence, our AQFT construction is canonical in these cases.
The aim of Section 6 is to analyze the dimensionally reduced AQFT B ± that is defined on the quotient 2-manifold B ± := M/ ∼ ± .Upon restriction to the interior IntB ± ⊆ B ± , we find that this AQFT specializes to the 2-dimensional part of linear Chern-Simons theory, which is given by canonical quantization of the Chern-Simons phase space along the Atiyah-Bott Poisson structure [AB83].Furthermore, restricting the AQFT B ± to a tubular neighborhood ∂B ± × [0, 1) ⊆ B ± of the boundary, we obtain an AQFT that is equivalent to the chiral free boson on the 1-manifold ∂B ± , see Corollary 6.3.The interplay between bulk and boundary degrees of freedom is realized in our approach through dg-algebra morphisms B ± (V ) → B ± (V ) that are associated with subset inclusions V ⊆ V ⊆ B ± from interior regions V ⊆ IntB ± into regions V ⊆ B ± that intersect the boundary, i.e.V ∩ ∂B ± = ∅.This allows us to describe certain Chern-Simons observables in the bulk in terms of equivalent chiral boson observables on the boundary, which is illustrated in Example 6.4 by studying the boundary observable corresponding to the Chern-Simons observable that measures the holonomy of flat connections.Appendix A contains some technical details that are required in the main part of the paper.
Notation and conventions for cochain complexes: Let us fix a field K of characteristic 0, e.g. the real numbers R or the complex numbers C. A cochain complex V consists a family of K-vector spaces {V i } i∈Z and a family of degree increasing linear maps {d : V i → V i+1 } i∈Z (called the differential) that squares to zero d 2 = 0.A cochain map f : V → W between two cochain complexes is a family of linear maps {f : V i → W i } i∈Z that commutes with the differentials f • d = d • f .We denote by Ch K the category of cochain complexes and cochain maps.
Given any cochain complex V and integer p ∈ Z, we define the p-shifted cochain complex V [p] by V [p] i := V i+p , for all i ∈ Z, and the differential d [p] := (−1) p d.
To each cochain complex V one can assign its cohomology, which is the graded vector space defined by H i (V ) := Ker(d : The category Ch K is closed symmetric monoidal.The monoidal product V ⊗ W of two cochain complexes is given by the family of vector spaces for all i ∈ Z, and the differential d ⊗ that is defined by the graded Leibniz rule for all homogeneous v ∈ V and w ∈ W , where | • | ∈ Z denotes the degree.The monoidal unit is K, regarded as a cochain complex concentrated in degree 0 with trivial differential, and the symmetric braiding given by the Koszul sign rule.
The internal hom [V, W ] between two cochain complexes is given by the family of vector spaces for all i ∈ Z, where Hom K denotes the vector space of linear maps, and the differential ∂ that is defined by the graded commutator Observe that a cochain map f : V → W is the same datum as a 0-cocycle in the internal hom complex [V, W ], i.e. f ∈ [V, W ] 0 such that ∂f = 0. Furthermore, a cochain homotopy between two cochain maps f, g : Higher cochain homotopies admit a similar interpretation in terms of the internal hom complex.

Chern-Simons with (anti-)self-dual boundary condition
In this section we describe the cochain complex that encodes linear Chern-Simons theory on an oriented 3-dimensional manifold M and its (−1)-shifted symplectic structure.The case where the boundary ∂M = ∅ is empty is quite standard and details can be found e.g. in the book of Costello and Gwilliam [CG17, Chapter 4.5].The case where the boundary ∂M = ∅ is non-empty is more involved as it requires the choice of a boundary condition to obtain a well-defined (−1)-shifted symplectic structure.These aspects, including particular choices of boundary conditions, have been studied systematically in the recent work [GRW22] of Gwilliam, Rabinovich and Williams, which serves as our main reference for this section.
Empty boundary ∂M = ∅: Consider an oriented 3-manifold M without boundary ∂M = ∅.The Chern-Simons complex is defined as the 1-shifted de Rham complex where d denotes the de Rham differential and round brackets indicate the cohomological degree.
It is important to note that the latter differs by 1 from the de Rham degree, i.e. |α| dR = |α| + 1 for all homogeneous α ∈ F(M ).The physical interpretation of the complex F(M ) is as follows: An element A ∈ F(M ) 0 = Ω1 (M ) in degree 0 is a gauge field (principal R-connection) and an element c ∈ F(M ) −1 = Ω 0 (M ) in degree −1 is a ghost field (infinitesimal gauge transformation).The elements in positive degrees are the antifields A ‡ ∈ F(M ) 1 = Ω 2 (M ) and the antifields for ghosts c ‡ ∈ F(M ) 2 = Ω 3 (M ).As typical for the BV formalism, the differential encodes both the action of gauge symmetries and the equation of motion, which in the case of linear Chern-Simons theory is the flatness condition dA = 0.
Using the wedge product of differential forms, we can define a cochain map to the 2-shifted de Rham complex.Observe that this map is graded antisymmetric, i.e.
for all homogeneous α, β ∈ F(M ).Since by hypothesis the boundary ∂M = ∅ is empty, integration of compactly supported forms defines a cochain map , which allows us to define a (−1)-shifted symplectic structure on the subcomplex F c (M ) ⊆ F(M ) of compactly supported sections.
Non-empty boundary ∂M = ∅: Consider an oriented 3-manifold M with non-empty boundary ∂M = ∅.As field complex we take again the 1-shifted de Rham complex (2.1), but now on a manifold with boundary.The (−1)-shifted symplectic structure (2.4) does not directly generalize to the present case because the map ω (−1) fails to be a cochain map.Indeed, as a consequence of Stokes' theorem, one has that for all homogeneous α, β ∈ F c (M ), where ι * denotes the pullback of differential forms along the boundary inclusion ι : ∂M → M .
One way to resolve this incompatibility between the differential and ω (−1) is to impose a suitable boundary condition on the field complex F(M ).A particularly interesting approach to boundary conditions, which draws inspiration from the intersection theory of Lagrangians in derived algebraic geometry [PTVV13], has been recently developed in [GRW22] and [Rab20;Rab21].Without going into the technical details, which are nicely explained in these papers, the key idea of this approach is to introduce a cochain complex F L (M ) of boundary conditioned fields by forming a fiber product in the category of cochain complexes. 1Let us explain the ingredients of this diagram, as well as their interpretation, in the context of our example: The boundary field complex F(∂M ) is the 1-shifted de Rham complex of the boundary manifold ∂M , i.e.

F(∂M
and the cochain map ι * is given by pullback of differential forms along ι : ∂M → M .Using an analogous construction as in (2.4), together with the fact that ∂M is 2-dimensional, one obtains an unshifted symplectic structure on the subcomplex F c (∂M ) ⊆ F(∂M ) of compactly supported sections.This allows us to rewrite Eqn.(2.5) in the following more conceptual form which means that ω (−1) defines a derived isotropic structure for the cochain map ι * in (2.6).As a consequence of the non-degeneracy condition proven in [Rab20, Lemma 2.20], ω (−1) is further (a field theoretic analogue of) a derived Lagrangian structure on ι * .
The subcomplex L ⊆ F(∂M ) in (2.6), which we call the boundary condition, is the choice of a local and strict Lagrangian in F(∂M ), ω (0) .Local means that L = Γ ∞ (L) is given by the sections of a subbundle L → ∂M of the boundary field bundle ∧ • T * ∂M [1] → ∂M , and strict Lagrangian means that L ⊆ F(∂M ) is strictly isotropic, i.e. the restriction of the unshifted symplectic structure to the subcomplex L c ⊆ L of compactly supported sections vanishes, and maximal in the sense that the total rank of L is half that of An explicit model for the cochain complex F L (M ) defined by the fiber product in (2.6) is given by the subcomplex F L (M ) ⊆ F(M ) of the field complex consisting of all fields α ∈ F(M ) that satisfy the boundary condition ι * (α) ∈ L. We denote the individual components of this subcomplex by (2) (2.11) Due to the strict isotropy condition (2.10) for L, one immediately observes that the boundary terms in (2.5) and (2.9) vanish for fields that satisfy the boundary condition, hence ω (−1) restricts to a (−1)-shifted symplectic structure (2.12) on the subcomplex F L,c (M ) ⊆ F L (M ) of compactly supported sections.
Note that, when the boundary ∂M = ∅ is empty, the boundary field complex F(∂M ) = 0 is trivial and so is any boundary condition L = 0. Hence, the cochain complex F L (M ) and its (−1)-shifted symplectic structure ω (−1) specialize to the ones we have described in the previous paragraph.
Linear observables and (−1)-shifted Poisson structure: We describe the linear observables for the complex F L (M ) of boundary conditioned fields by the 1-shifted cochain complex of compactly supported sections.The evaluation of linear observables on fields is given by the cochain map ev : that is obtained from (2.2) and the integration map.Explicitly, for all homogeneous ϕ ∈ F L,c (M )[1] and α ∈ F L (M ).The complex of linear observables can be endowed with the (−1)-shifted Poisson structure where the first cochain isomorphism is constructed from the symmetric braiding on the category of cochain complexes, i.e. it involves Koszul signs.These Koszul signs cancel the minus signs entering ω (−1) through (2.2), such that is simply given by the forming the wedge product followed by integration over M , for all linear observables ϕ, ψ ∈ F L,c (M ) [1].
(Anti-)self-dual boundary condition: Of particular interest to us are Lorentzian geometric variants of the so-called chiral and anti-chiral WZW boundary conditions, which were introduced in [GRW22, Example 2.3] through the choice of a complex structure on ∂M and the corresponding Dolbeault decomposition of Ω • (∂M ).To describe a Lorentzian analogue of these boundary conditions, we assume that ∂M comes endowed with (the conformal class of) a Lorentzian metric g and a time orientation.As orientation on ∂M we take the one induced from the orientation of the bulk M .From these data we can define a Hodge operator * : Ω k (∂M ) → Ω 2−k (∂M ), which due to the Lorentzian signature of the metric squares to the identity * 2 = id on 1-forms.This allows us to decompose the vector space of 1-forms Introducing the standard light cone coordinates x ± := t ± x, the Hodge operator acts on dx ± ∈ Ω 1 (∂M ) according to * (dx ± ) = ∓dx ± , i.e. dx − defines a module basis for the self-dual forms Ω + (∂M ) and dx + defines one for the anti-self-dual forms Ω − (∂M ).Note the flip of sign.
We define the self-dual boundary condition as the subcomplex of the boundary field complex (2.7).The anti-self-dual boundary condition L − is similarly defined with Ω − (∂M ) in place of Ω + (∂M ), i.e.
Throughout the whole paper, we impose either L + or L − as boundary condition on the Chern-Simons theory on M .2For a uniform notation, we denote the chosen boundary condition as L ± .The corresponding boundary conditioned fields introduced in (2.6) then satisfy the following boundary conditions: The pullback to the boundary ι * (c) = 0 of the ghost field vanishes and the pullback to the boundary ι * (A) ∈ Ω ± (∂M ) of the gauge field is an (anti-)self-dual 1-form.Note that the antifields A ‡ and c ‡ are not restricted by the (anti-)self-dual boundary condition.

Construction of Green's homotopies
The data presented in Section 2 is sufficient to quantize linear Chern-Simons theory with the chosen (anti-)self-dual boundary condition in terms of a factorization algebra.The relevant construction is given by linear BV quantization of the complex of linear observables (2.13) along its (−1)-shifted Poisson structure (2.15), see e.g.[CG17, Chapter 4] or [GRW22, Section 3] for the details.
In the present paper we shall take a different approach and study the quantization of this theory in terms of an AQFT.This requires additional structure, given by so-called Green's homotopies [BMS22a], which generalize the concept of retarded/advanced Green's operators to a homotopical context.Using these Green's homotopies, one can determine as in [BMS22a] an unshifted Poisson structure on the complex of linear observables (2.13), which then can be quantized by constructing canonical commutation relation (CCR) dg-algebras as in [BBS20].
Example 3.1.Green's homotopies provide a natural homotopical generalization of the usual retarded/advanced Green's operators for normally hyperbolic partial differential equations.To illustrate the main idea, let us briefly sketch the concept of Green's homotopies in the simplest example given by Klein-Gordon theory on a globally hyperbolic Lorentzian manifold N .We refer the reader to [BMS22a] and also [BBS20, Section 4.1] for more details.The field complex of the Klein-Gordon field is given by where denotes the d'Alembertian and m 2 ≥ 0 is a mass term.Elements Φ ∈ Ω 0 (N ) in degree 0 are interpreted as fields and elements Φ ‡ ∈ Ω 0 (N ) in degree 1 as antifields.Note that the cohomology of this complex is concentrated in degree 0 and it is given by the usual solution space of the Klein-Gordon equation A retarded/advanced Green's operator for Klein-Gordon theory is defined as a linear map The first property can be easily described in the language of cochain complexes: The re-tarded/advanced Green's operator determines via −1 which defines a cochain homotopy ∂G ± = j from the zero map to the cochain map j : F KG c (N ) → F KG (N ) that includes compactly supported 0-forms into all 0-forms.The second property of Green's operators states that this homotopy must have suitable support properties, which can be formalized in a homotopically meaningful way as in [BMS22a].Taking the difference G −1 of the retarded and the advanced Green's homotopy defines a cochain map G : where vol N denotes the volume form.
The advantage of Green's homotopies is that they exist also in gauge-theoretic examples, which are described by complexes that encode also ghosts and their antifields, and hence are longer than (3.1), while ordinary Green's operators can not exist in a gauge theory (before gauge fixing) due to degeneracy of the equation of motion.
Our strategy in this section is as follows: In Subsection 3.1 we use relevant chiral geometry (in the sense of [Gra23]) of the boundary Lorentzian manifold ∂M to define a notion of Green's homotopies for the (anti-)self-dual boundary condition L ± in (2.18).A key feature of this chiral geometry is the so-called chiral flows.In Subsection 3.2 we abstract to Green's homotopies for the de Rham complex on a manifold that is endowed with a smooth R-action (i.e. a flow), and give an explicit construction of such Green's homotopies under suitable niceness conditions on the R-action.Finally, in Subsection 3.3 we specialize this construction to produce compatible Green's homotopies for the boundary condition L ± and the complex F L ± (M ) of boundary conditioned fields, under the hypothesis that the bulk M may be equipped with suitable extensions of a chiral flow on ∂M .

Chiral geometry of the boundary
To motivate our notion of Green's homotopies for Chern-Simons theory subject to an (anti-)selfdual boundary condition, we first consider the analogous notion of Green's homotopies for the boundary condition complex L ± itself.In so doing, we will encounter relevant chiral features of the 2-dimensional boundary spacetime ∂M , which will inform our construction in the bulk M .
In this subsection we will work on the 2-dimensional boundary manifold ∂M , which by our hypotheses is endowed with a Lorentzian metric g, an orientation o ∂M and a time-orientation t.
In the following definition we represent the latter two data by a non-vanishing 2-form ω ∈ Ω 2 (∂M ) and a non-vanishing time-like vector field τ ∈ Γ ∞ (T ∂M ).
(b) A null curve γ : [0, 1] → ∂M is called -pointing if all its tangent vectors are -pointing.The other three cases , and are defined similarly.
Evidently, any -pointing curve γ produces a -pointing curve s → γ(1 − s) by reversing its parameterization, and vice versa.-and -pointing curves are similarly related.Remark 3.4.In [Gra23, Section 4], a null curve γ : [0, 1] → ∂M that is either -pointing or -pointing is called right-chiral, and one that is either -pointing or -pointing is called left-chiral.In the present paper we prefer to use the finer distinction from Definition 3.2.
Our concept of pointings for tangent vectors is related to (anti-)self-duality of 1-forms.
Lemma 3.5.A 1-form α ∈ Ω 1 (∂M ) is self-dual if and only if, at every point p ∈ ∂M , the contraction α(v) = 0 vanishes for all -pointing and all -pointing null vectors v ∈ T p ∂M .Similarly, a 1-form α ∈ Ω 1 (∂M ) is anti-self-dual if and only if, at every point p ∈ ∂M , the contraction α(v) = 0 vanishes for all -pointing and all -pointing null vectors v ∈ T p ∂M .
Proof.Since every 2-dimensional Lorentzian manifold is locally conformally equivalent to the Minkowski spacetime, and since (anti-)self-duality and the pointings from Definition 3.2 depend only on the conformal structure, it is sufficient to prove the claim for the Minkowski spacetime.The latter follows immediately from our explicit descriptions in Examples 2.1 and 3.3.
Making use of the pointings from Definition 3.2, we can introduce for each point p ∈ ∂M the following subset and similar subsets for the other three cases , and .These subsets characterize the four null components of the light cone at p ∈ ∂M .Given any subset S ⊆ ∂M , we further define and similar in the other three cases.
Remark 3.6.Since reversing the parameterization changes a -pointing curve into a -pointing curve, it follows that p ∈ J (q) if and only if q ∈ J (p).In fact, J (p) and J (p) arise respectively as the sets of successors and predecessors of p under the binary relation ≤ + on ∂M defined by p ≤ + q if either p = q or there exists a -pointing curve in ∂M from p to q.Similarly, J (p) and J (p) arise respectively as the sets of successors and predecessors of p under the binary relation ≤ − on ∂M defined by p ≤ − q if either p = q or there exists a -pointing curve in ∂M from p to q.
In the next definition we introduce the variants of Green's homotopies that are relevant for the present paper.See also Remark 3.8 below for some additional comments on the relationship to the framework of Green's homotopies developed in [BMS22a].
Definition 3.7.A -pointed Green's homotopy for the boundary condition −1 in the internal hom complex from the complex of compactly supported sections L ± c to L ± that satisfies the following properties: (i) ∂G = j, where j : L ± c → L ± is the canonical inclusion cochain map from compactly supported sections to all sections; (ii) supp G α ⊆ J supp α , for all α ∈ L ± c .The definition of Green's homotopies for the other three pointings , and is similar.
Remark 3.8.The following remarks are in order: 1.There are four types of Green's homotopies referring to the four different components of the light cone in 2-dimensional conformal Lorentzian geometry.These refine the two types of Green's homotopies (retarded and advanced) in standard m-dimensional Lorentzian geometry.
2. The definition of Green's homotopies in [BMS22a, Definition 3.5] is more general as it allows for pseudo-naturality under inclusions K ⊆ K of compact subsets of ∂M .From a conceptual perspective, this generality is needed to prove a uniqueness theorem for Green's homotopies stating that the space of Green's homotopies is either empty or contractible, see [BMS22a, Proposition 3.9].This level of generality is not needed in the present paper since all of our Green's homotopies will be of the strict type as defined in Definition 3.7.
Later in Proposition 3.22 we will show that, under suitable chiral niceness properties of ∂M , the self-dual boundary condition L + admits -and -pointed Green's homotopies and the anti-self-dual boundary condition L − admits -and -pointed ones.
As a preparation for our construction of Green's homotopies, let us recall from [Gra23, Proposition 4.14] that on each oriented and time-oriented Lorentzian 2-manifold ∂M there exists a so-called chiral frame (n − , n + ) for the tangent bundle T ∂M , i.e. n − ∈ Γ ∞ (T ∂M ) is an everywhere -pointing null vector field and n + ∈ Γ ∞ (T ∂M ) is an everywhere -pointing vector field.The choice of chiral frame is canonical up to rescalings n − ∼ f − n − and n + ∼ f + n + by positive smooth functions f ∓ ∈ C ∞ (∂M, R >0 ).Furthermore, by [Gra23,Lemma 4.19], one may assume without loss of generality that both n − and n + are complete vector fields.This allows us to define the +-and −-chiral flows as the global flows of the complete vector fields n + and n − respectively.
Chiral flows provide a useful technical tool to handle the variously-pointed null curves of Definition 3.2.By [Gra23, Lemma 4.21], any -or -pointing null curve in ∂M is, up to reparameterization, an integral curve associated to the +-chiral flow Θ + .Similarly, the -and -pointing null curves are described by the −-chiral flow Θ − .
Remark 3.10.There exists no canonical choice of the chiral flows Θ + and Θ − on ∂M , owing to the freedom to rescale n + and n − by positive smooth functions.However, the orbits of any two +-chiral flows Θ + and Θ + coincide, being the images of inextendable -pointing null curves in ∂M .Thus, the quotient π + : ∂M → ∂M/ ∼ + is canonical on a given 2-dimensional spacetime ∂M , for ∼ + the orbit relation of any +-chiral flow.Similarly, the quotient π − : ∂M → ∂M/ ∼ − by any −-chiral flow is canonical.
Remark 3.11.One may describe the sets J (p) and J (p) from (3.4a) in terms of the +-chiral flow only.Indeed, by [Gra23, Proposition 4.22], a point q ∈ ∂M lies in J (p) (or equivalently, p ∈ ∂M lies in J (q)) if and only if q = Θ + (s, p) for some s ≥ 0. The sets J (p) and J (p) may be characterized similarly using the −-chiral flow.
Just as we denote by L ± our choice of self-dual or anti-self-dual boundary condition, we henceforth denote by Θ ± a corresponding choice of +-or −-chiral flow on ∂M (+-chiral if the chosen boundary condition is self-dual, −-chiral if the boundary condition is anti-self-dual).

Green's homotopies associated with a proper R-action
Our construction of Green's homotopies for L ± will use only the chiral flow Θ ± on ∂M , without directly referring to any of the other available data, e.g. the Lorentzian metric, orientation or timeorientation.To simplify notation during this construction, we temporarily abstract to the k-shifted de Rham complex Ω • (N )[k] of an arbitrary smooth manifold N (with or without boundary) that is endowed with a smooth R-action (i.e.flow) Θ : R × N → N .Our general construction can be specialized in the case of (N, Θ) = (∂M, Θ ± ) to provide appropriately-pointed Green's homotopies for L ± ⊆ Ω • (∂M ) [1].This abstraction will also be of use later in Proposition 3.28, when we will take N = M to build appropriate Green's homotopies for the complex Given any manifold N that is endowed with an R-action Θ : R × N → N , we can define analogues of the subsets J (S) from (3.4) and their , , variants.We define the Θ-future of a point p ∈ N as the subset and the Θ-past of p ∈ N as the subset Recalling Remark 3.11, we observe that J ↑ Θ + (p) = J (p) and J ↓ Θ + (p) = J (p) for Θ + a +-chiral flow on N = ∂M , and J ↑ Θ − (p) = J (p) and J ↓ Θ − (p) = J (p) for Θ − a −-chiral flow.We also define for any subset S ⊆ N the Θ-future/past of S to be (3.7) With this we can generalize Definition 3.7 of Green's homotopies for L ± to the k-shifted de Rham complex Ω • (N )[k] on N .
Definition 3.12.Let N be a smooth manifold (with or without boundary) that is endowed with an R-action Θ : R × N → N .A forward/backward Green's homotopy for the k-shifted de Rham complex Ω −1 in the internal hom complex that satisfies the following properties: is the canonical inclusion cochain map from compactly supported forms to all forms; Remark 3.13.Note that a forward/backward Green's homotopy −1 for the unshifted de Rham complex induces corresponding forward/backward Green's homotopies −1 for each k-shifted de Rham complex.The sign factor ensures that property (i), i.e. ∂G

↑/↓
[k] = j, holds true for the internal hom differential ∂ associated with the k-shifted de Rham differential d [k] = (−1) k d.For this reason, we construct our candidate Green's homotopies below only for the unshifted de Rham complex.
Let us consider the following span of smooth maps where pr denotes the projection onto the second factor.We will construct Green's homotopies in the sense of Definition 3.12 by first pulling forms back along Θ and then integrating over (suitable subsets of) the fibers of pr.The latter is a trivial bundle with oriented fiber, and so is oriented as a bundle.To ensure that the composite operation is well-defined on compactly supported forms, we have to impose a niceness condition on the R-action Θ, namely that it is proper.Let us start by recalling that, for a smooth fiber bundle π : E → B with fibers of dimension r, fiber integration π * : with vertically compact support.This support condition means that, given any compact subset K ⊆ B in the base, the subset π −1 (K) ∩ supp α ⊆ E is compact in the total space.For standard references on fiber integration, see e.
is a closed subset.Using now that Θ is a proper R-action, i.e. its shear map θ is a proper map, it follows that pr −1 (K) ∩ supp(Θ * α) ⊆ θ −1 K × supp α is a closed subset of a compact set, and hence is compact.
To propose a candidate for a forward Green's homotopy, we restrict the R-action to non-positive parameters R ≤0 ⊆ R and consider the corresponding restriction . By an obvious corollary of Lemma 3.14, if Θ is a proper R-action, we obtain a family of linear maps by composing the pullback of forms along Θ with the fiber integration R ≤0 := pr * along pr.These components define the (−1)-cochain Restricting to non-negative parameters, we obtain again a restricted span Taking the additive inverses of these components defines the (−1)-cochain Beyond well-definedness of (3.12) and (3.14), the assumption that Θ is proper gives some further technical consequences described in Appendix A. These enter the proof of the following Proposition 3.15.Suppose that the R-action Θ : R × N → N is proper.Then G ↑ Θ in (3.12) defines a forward Green's homotopy and G ↓ Θ in (3.14) defines a backward Green's homotopy.
Proof.We have to verify the two properties (i) and (ii) from Definition 3.12.
Item (i) is a simple consequence of the fiber-wise Stokes theorem, see e.g.[Nic21, Theorem 3.4.54].For the (−1)-cochain where in the second step we used that the de Rham differential commutes with pullbacks of differential forms and in the third step we used the fiber-wise Stokes theorem.Since the boundary ∂R ≤0 = {0} is a singleton, the fiber integration ∂R ≤0 is given by the pullback of forms along N → R ≤0 × N , p → (0, p).Since Θ(0, • ) = id N yields the identity, this shows that is the canonical inclusion.The proof for ∂G ↓ Θ = j is similar and uses that the boundary ∂R ≥0 = {0} is negatively oriented, which accounts for the minus sign in (3.14).
To prove item (ii), let us first note that the subset J ↑/↓ Θ (K) ⊆ N is closed for each compact subset K ⊆ N because the R-action Θ is proper, see Corollary A.7.Let us take any compactly supported form α ∈ Ω k c (N ) and consider G ↑ Θ α ∈ Ω k−1 (N ).Recall that supp G ↑ Θ α ⊆ N is defined as the closure of the set of points p ∈ N for which the evaluation G ↑ Θ α (p) = 0 is non-vanishing.It suffices to show that each such point satisfies p ∈ J ↑ Θ supp α as the latter set is closed.From G ↑ Θ α (p) = R ≤0 Θ * α (p) = 0, it follows that there exists a parameter s ≤ 0 such that Θ * α (s, p) = 0, hence one finds a point (s, p) ∈ supp(Θ * α) ⊆ Θ −1 (supp α).Since Θ(s, p) ∈ supp α for non-positive s ≤ 0, we conclude that p ∈ J ↑ Θ supp α .By a similar argument one shows that supp as restricted fiber integrations: Our Green's homotopies ] −1 may be understood as integration over "backward"/"forward" halves of the fibers of the quotient π : N → N/R of N by the R-action Θ.The latter is a principal R-bundle by Corollary A.2 because the R-action Θ is proper.We now demonstrate this interpretation using appropriate coordinate expressions.
Denote B := N/R.We work in a trivialization N ∼ = R × B, which exists since principal Rbundles are trivializable owing to the contractibility of R. Forms on the product space decompose (3.17) Since the fiber integrations in and similarly Recall that the fiber integration π * : Ω • vc (N ) → Ω •−1 (B) over the bundle π : N → B (canonically oriented as per Lemma A.3) has a similar expression where π : N → N/R is the quotient by the proper R-action Θ.

Boundary and bulk Green's homotopies
We now specialize the Green's homotopies from Subsection 3.2, first to the self-dual and anti-selfdual boundary condition complexes L + and L − and finally to the complex F L ± (M ) of boundary conditioned fields of linear Chern-Simons theory on M .
Boundary Green's homotopies: For our construction of the Green's homotopies (3.12) and (3.14) for the de Rham complex, it was crucial to assume that the R-action is proper.Since the chiral flows Θ + , Θ − : R × ∂M → ∂M on a 2-dimensional oriented and time-oriented Lorentzian manifold ∂M are in general not proper, we have to assume the following strong chiral niceness condition.
Remark 3.18.If Θ ± is proper, then the associated quotient map π ± : ∂M → ∂M/ ∼ ± has the structure of a smooth principal R-bundle, see Corollary A.2.Because R is contractible, the quotient map π ± admits smooth global sections, which can be used to define a chiral analogue of Cauchy surfaces in ∂M [Gra23, Section 4.2].Properness of the chosen chiral flow Θ ± may therefore be understood as a chiral analogue of global hyperbolicity.
Example 3.19.If ∂M is globally hyperbolic (in the usual sense), then any +-or −-chiral flow on it is proper.Hence, Assumption 3.17 holds true whenever ∂M is globally hyperbolic.This gives a large class of examples satisfying our hypotheses.
In order to prove that the Green's homotopies from Proposition 3.15 restrict to the boundary condition L ± ⊆ Ω • (∂M )[1] we require the following technical lemma.
Lemma 3.20.Suppose that Θ + : R × ∂M → ∂M is a proper +-chiral flow.Then there exists a nowhere-vanishing self-dual form β + ∈ Ω + (∂M ) that is invariant under Θ + in the sense that where pr : R × ∂M → ∂M denotes the projection map.
Proposition 3.22.Suppose that ∂M has a proper +-chiral flow Θ + .Then the self-dual boundary condition L + admits a -pointed Green's homotopy G and a -pointed Green's homotopy G in the sense of Definition 3.7.An explicit choice is given by restricting the Green's homotopies from Proposition 3.15 and Remark 3.13 to the boundary condition Similarly, if ∂M has a proper −-chiral flow Θ − then the anti-self-dual boundary condition L − admits a -pointed Green's homotopy G and a -pointed Green's homotopy G , with an explicit choice given by restricting Proof.The only nontrivial step is to show that the restrictions exist.We only spell out the proof for the case of L + and G ↑ Θ + [1] since all other cases are similar.Recalling the explicit form of the boundary condition complex (2.18), we have to show that, for every compactly supported self-dual 1-form α ∈ Ω + c (∂M ), the 0-form −G ↑ Θ + (α) ∈ Ω 0 (∂M ) is zero and that, for every compactly supported 2-form α ∈ Ω 2 c (∂M ), the 1-form −G ↑ Θ + (α) ∈ Ω 1 (∂M ) is self-dual.Using the Θ + -invariant module basis from Lemma 3.20, we can write α = φ ∧ β + where φ ∈ C ∞ c (∂M ) is a compactly supported function in the first case and φ ∈ Ω − c (∂M ) is a compactly supported anti-self-dual form in the second case.Using (3.12), we compute where in the second step we used that β + is Θ + -invariant (see Lemma 3.20) and the last step is an application of the projection formula for fiber integration [Nic21, Equation (3.4.11)].In the first case φ is a 0-form, so its fiber integral over 1-dimensional fibers vanishes.It follows that −G ↑ Θ + (α) = 0.In the second case φ is a 1-form, so its fiber integral is a function.Since Bulk Green's homotopies: We would like to extend the boundary Green's homotopies from Proposition 3.22 to the complex F L ± (M ) ⊆ Ω • (M )[1] of boundary conditioned Chern-Simons fields on M .Such extensions can be obtained from the techniques developed in Subsection 3.2, provided that the following assumption holds true.
Assumption 3.23.The bulk manifold M admits a proper R-action Θ ± : R × M → M that extends the chosen chiral flow Θ ± on the boundary ∂M , i.e. the diagram commutes, where ι : ∂M → M denotes the boundary inclusion.
Remark 3.24.The extension Θ ± of the chosen chiral flow Θ ± required by Assumption 3.23 is clearly not unique.In Section 4, we will construct an AQFT for the Chern-Simons/Wess-Zumino-Witten bulk/boundary system out of Θ ± and the associated Green's homotopies from Proposition 3.28 below.This prompts the question: How canonical is our AQFT construction?
As we will show in Section 5, the AQFT for this bulk/boundary system is (up to weak equivalence) only sensitive to the base manifold of the quotient map π ± : M → M/ ∼ ± associated with the R-action Θ ± .Under certain additional topological assumptions on M , we show in Proposition 5.6 that any two such base manifolds arising from different choices of Θ ± must be diffeomorphic.From this, it follows that any two proper extensions of the chiral flow Θ ± to M yield weakly equivalent AQFTs.
Remark 3.25.Commutativity of (3.25) implies in particular that ∂M is a Θ ± -invariant subspace of M .It is a general fact of continuous group actions that if Θ ± is proper, then its restriction to any invariant subspace is also proper.Consequently, Assumption 3.23 implies Assumption 3.17 that the chosen chiral flow Θ ± on the boundary is proper.
Example 3.26.Continuing Examples 2.1, 3.3 and 3.9, consider M = R 2 × R ≥0 with boundary given by the standard Minkowski spacetime.Then the proper R-action provides an extension of the chiral flows Θ + and Θ − from Example 3.9 when = +1 and = −1 respectively.
Proposition 3.28.Suppose that Assumption 3.23 holds.Then the forward/backward Green's homotopies from Proposition 3.15 and Remark 3.13 restrict to the complex of boundary conditioned fields.These bulk Green's homotopies are compatible with the boundary ones from Proposition 3.22 in the sense that where ι : ∂M → M denotes the boundary inclusion.
Proof.Recalling the definition of ), we observe that the compatibility conditions (3.29) together with Proposition 3.22 would imply that the desired restrictions exist.Indeed, we would have to show that ι * G which is fiber-wise a diffeomorphism.Recalling (3.12), we compute where in the second step we used [GHV72, Section 7.12, Proposition VIII] and in the third step we used the commutative diagram (3.25).

The AQFT for the bulk/boundary system
For M an oriented 3-manifold whose boundary ∂M is endowed with a Lorentzian metric and a time-orientation, we have defined in (2.18) a boundary condition L ± for linear Chern-Simons theory on M .This boundary condition is associated to a choice of self-dual or anti-self-dual forms on ∂M , and in turn to a choice of +-or −-chiral flow Θ ± .In Assumption 3.23, we have assumed the existence, and made a choice, of a proper R-action Θ ± : R × M → M on the whole of M , which restricts to the chosen chiral flow Θ ± on the boundary.In the preceding section, we demonstrated the existence of Green's homotopies associated to the chosen R-action Θ ± .
The goal of this section is to describe a suitable type of AQFT on M that quantizes the boundary conditioned Chern-Simons fields F L ± (M ) from (2.6) for the chosen (anti-)self-dual boundary condition L ± .
The relevant orthogonal category: Recall that an AQFT can be defined in the very broad context where an orthogonal category [BSW21; BSW19] is used to model a category of "spacetime regions" with a notion of "independence" of such.The prime example for relativistic AQFT is the category COpen(N ) of all non-empty causally convex open subsets U ⊆ N of an oriented and time-oriented globally hyperbolic Lorentzian manifold N , with orthogonality relation determined by causal disjointness.However, this choice is not suitable for our boundary conditioned Chern-Simons theory, because the bulk M does not come equipped with a Lorentzian metric.Using our choice of proper R-action Θ ± : R × M → M that extends a chiral flow on the boundary ∂M , we propose the following more suitable category.Definition 4.1.We denote by Θ ± -Open(M ) the category whose objects are all non-empty open subsets U ⊆ M that are Θ ± -convex, in the sense that and whose morphisms ι U U : U → U are subset inclusions U ⊆ U ⊆ M .We endow this category with the orthogonality relation defined by Remark 4.2.The concepts of Θ ± -convexity and Θ ± -disjointness are analogues of causal convexity and causal disjointness in relativistic AQFT.It is sometimes useful to rephrase these definitions in the following equivalent ways.A subset U ⊆ M is Θ ± -convex if and only if the orbit of each point p ∈ U under the R-action Θ ± : R × M → M does not exit and re-enter U ⊆ M .Two subsets (U 1 ⊆ U ) ⊥ (U 2 ⊆ U ) are Θ ± -disjoint if and only if the orbit of U 1 under the R-action The concept of Cauchy morphisms from relativistic AQFT also finds an analogue in our context.Recall that one of the many equivalent characterizations for a morphism ι U U : U → U between two causally convex subsets U ⊆ U ⊆ N in an oriented and time-oriented globally hyperbolic Lorentzian manifold N to be Cauchy is to require that the Cauchy developments D(U ) = D(U ) ⊆ N in N agree.This easily generalizes to our framework.
Recalling also Remark 3.18, we may regard smooth global sections of the quotient map π ± : M → M/ ∼ ± (which is a smooth principal R-bundle by Corollary A.2) as appropriate analogues of Cauchy surfaces in our chiral geometric context.Then ι U U : U → U is a Θ ± -Cauchy morphism if and only if U contains (the image of) a smooth section of the restriction Informally, this says that a morphism is Θ ± -Cauchy if its source contains (the chiral analogue of) a Cauchy surface for its target.
An AQFT on this orthogonal category is then defined as a functor to the category of associative and unital differential graded * -algebras (in short, dg-algebras) over C that satisfies the Einstein causality axiom for ⊥: For each orthogonal pair (U of cochain maps commutes, where µ denotes the (opposite) multiplication on the dg-algebra A(U ) ∈ dg * Alg C .Such AQFT is said to satisfy the time-slice axiom if, for each Θ ± -Cauchy morphism ι U U : U → U in Θ ± -Open(M ), the cochain map underlying the dg * Alg C -morphism Poisson cochain complexes: Using the Green's homotopies from Proposition 3.28, we define the (−1)-cocycle which equivalently can be regarded as a cochain map from the 1-shift of the complex of compactly supported sections.(Recall that these are the linear observables from (2.13).)The role played by G Θ ± [1] is analogous to the causal propagator (or retarded-minus-advanced Green's operator) in relativistic AQFT.In particular, we can define, for each object U ∈ Θ ± -Open(M ), a cochain map where ext denotes extension by zero of compactly supported sections and ev was defined in (2.14) in terms of integration over M .Note that, by construction, this family of cochain maps is natural on the category Θ ± -Open(M ) in the sense that, given any morphism U ⊆ U in this category, we have that To ease notation, we shall often drop the superscript U and simply write τ (0) for every member of the family {τ U (0) } of cochain maps.To show that (4.8) defines indeed an unshifted linear Poisson structure, i.e. τ (0) is graded antisymmetric, we make use of the following result that allows us to rewrite this map in terms of an integral over the base manifold B ± of the quotient map π ± : M → M/ ∼ ± =: B ± .Recall that the latter is a smooth principal R-bundle by Corollary A.2, and that this induces on B ± an orientation that is appropriately compatible with the orientation of M , see Corollary A.4.
Lemma 4.5.The cochain map (4.8) can be written as Proof.From Proposition 3.16, we have From this equivalent description, it is immediate to see that the cochain maps (4.8) are graded antisymmetric, hence they define unshifted linear Poisson structures.Using the naturality observed in (4.9), we further obtain that the assignment U → F L ± ,c (U )[1], τ U (0) extends to a functor taking values in the category of Poisson cochain complexes.
Proposition 4.6.The functor (4.12) satisfies the following classical analogues of the Einstein causality and time-slice axioms: (a) For each orthogonal pair is a quasi-isomorphism.
Proof.Item (a) follows immediately from the support properties of Green's homotopies in Definition 3.12.(Recall that the map ev in (4.8) is given by integration over M , see (2.14).)The proof of item (b) is analogous to the proof of the corresponding statement in the relativistic case, see e.g.[BMS22b, Theorem 3.10].Where in the relativistic case one uses Cauchy surfaces to construct a quasi-inverse, we use sections of the quotient map π ± : M → B ± , recalling Remarks 3.18 and 4.4.The compactness property used in the relativistic case may be substituted by Lemma A.8.
The AQFT: The quantization of our bulk/boundary system is easy to carry out by using the cochain complex-valued canonical commutation relations functor CCR : PoCh R → dg * Alg C from [BBS20, Section 5].Post-composing the functor (4.12) with CCR defines the dg * Alg C -valued functor that assigns to U ∈ Θ ± -Open(M ) the dg-algebra constructed as the complexified tensor algebra over the cochain complex F L ± ,c (U )[1] of linear observables, modulo the two-sided differential graded * -ideal I τ U that is generated by the canonical commutation relations for all homogeneous generators ϕ, ψ ∈ F L ± ,c (U ) [1].The functorial structure is given by extending to dg-algebra morphisms the extension-by-zero maps ext.As a consequence of Proposition 4.6 and standard arguments, see e.g.[BBS20, Lemma 6.18], it follows that (4.15) satisfies the AQFT axioms.Let us record this result.

Dimensional reduction to the base manifold
As a consequence of the weak variant of the time-slice axiom (4.6) in terms of quasi-isomorphisms, which is sometimes called the homotopy time-slice axiom, it is difficult to analyze the AQFT A ± constructed in Corollary 4.7 directly.The goal of this section is to replace A ± by a weakly equivalent model B ± living on the base space of the quotient map π ± : M → M/ ∼ ± =: B ± associated with the proper R-action Θ ± : R × M → M .This base space B ± is an oriented, smooth 2-manifold, see Corollaries A.2 and A.4, so one achieves a dimensional reduction from the 3-dimensional manifold M to the 2-dimensional manifold B ± .Note that the base manifold B ± := M/ ∼ ± has a non-empty boundary ∂B ± = ∂M/ ∼ ± since ∂M is a Θ ± -invariant subspace of M as per (3.29).Hence, our dimensional reduction allows us to analyze boundary features of our bulk/boundary system by working in a simpler but equivalent 2-dimensional setup.
It is important to stress that this dimensional reduction is not to the boundary ∂M of M , and is therefore not itself exhibiting a CS/WZW correspondence.(The latter will be studied in Section 6 below.)The present dimensional reduction should be compared to the description of chiral CFTs by conformal nets.In that case, a chiral conformal theory on a 2-dimensional spacetime (say Minkowski R 1+1 or an appropriate compactification thereof) is equivalently described by data on a 1-dimensional light ray in that spacetime.This is possible because the fields in a chiral theory are independent of the position along the transverse light ray.The analogous independence exploited by our dimensional reduction is exactly that expressed by the time-slice axiom (4.6).
To construct this dimensionally reduced model, we begin by replacing A ± by a weakly equivalent model that satisfies the time-slice axiom strictly in terms of isomorphisms.The relevant mathematical framework for such strictification constructions has been developed in [BCS23], in the context of model category theory.

Orthogonal localization:
We describe an explicit model for the localization of the orthogonal category Θ ± -Open(M ) from Section 4 at all Θ ± -Cauchy morphisms.We also will show that this localization is reflective, hence [BCS23, Theorem 3.6] provides the desired strictification construction.Let us start by defining the orthogonal categories that we will identify later as the desired localizations.
Definition 5.1.We denote by Open(B ± ) the category whose objects are all non-empty open subsets V ⊆ B ± of the base manifold of the quotient map π ± : M → M/ ∼ ± =: B ± and whose morphisms ι V V : V → V are subset inclusions V ⊆ V ⊆ B ± .We endow this category with the disjointness orthogonality relation, i.e.
Taking preimages under π ± : M → B ± defines a functor to the category introduced in Definition 4.1 since the preimage π −1 ± (V ) ⊆ M of any non-empty open subset V ⊆ B ± is non-empty, open and Θ ± -convex.This functor preserves the orthogonality relations on the respective categories, i.e. it defines an orthogonal functor in the sense of [BSW21;BCS23].Since the fiber bundle projection π ± : M → B ± is an open map, we can also take images and define another functor that goes in the opposite direction.Also this functor is easily seen to preserve the orthogonality relations.We now observe that these two (orthogonal) functors define an adjunction π ± π −1 ± with counit : π ± π −1 ± ⇒ id defined component-wise by the identity morphisms Proposition 5.2.The functor (5.2) defines an orthogonal localization of Θ ± -Open(M ) at the set of all Θ ± -Cauchy morphisms.This orthogonal localization is reflective with reflector (5.1) ) is Θ ± -Cauchy according to Definition 4.3 exactly when the functor π ± sends it to an identity morphism.Since the right adjoint functor (5.1) is fully faithful, the statement then follows from [BCS23, Proposition 3.3] and the fact that the only isomorphisms in Open(B ± ) are identities.
From [BCS23, Theorem 3.6], we obtain an AQFT A st ± : Θ ± -Open(M ) → dg * Alg C that is weakly equivalent to the AQFT A ± from Corollary 4.7 but has the technical advantage that it satisfies the strict time-slice axiom, i.e. to any Θ ± -Cauchy morphism ι U in contrast to a quasi-isomorphism.Using also [BCS23, Corollary 3.7], one can construct this AQFT very explicitly as the pullback A st ± = ( π ± ) * ( π −1 ± ) * (A ± ) of the original A ± along the two orthogonal functors (5.1) and (5.2).Observe that the intermediate AQFT ( π −1 ± ) * (A ± ) : Open(B ± ) → dg * Alg C in this construction is defined on the base manifold B ± .By [BCS23, Theorem 3.6], this AQFT carries equivalent information to A ± , and hence also to its strictification A st ± , which means that we have achieved a dimensional reduction from the 3-manifold M to the 2-dimensional base manifold B ± .
Simplification: Even though the AQFT ( π −1 ± ) * (A ± ) : Open(B ± ) → dg * Alg C is defined on the category of non-empty opens of the 2-dimensional manifold B ± , the dg-algebras . The aim of this paragraph is to construct a weakly equivalent model for the AQFT ( π −1 ± ) * (A ± ) that is determined only from 2-dimensional data on B ± .Inspired by Lemma 4.5, we consider the fiber integration Lemma 5.3.The cochain map (5.4) factors through the subcomplex where Ω 0 ∂,c (V ) ⊆ Ω 0 c (V ) denotes the subspace of all compactly supported 0-forms on V whose pullback to the boundary ∂V vanishes.(In the case of ∂V = ∅, this boundary condition becomes void.)Moreover, the corestriction Proof.To simplify this proof, we pick a trivialization and denote by τ a choice of global coordinate on R. The vector space of (compactly supported) k-forms on the product manifold R × V admits a decomposition into forms that do have a leg along R and forms that do not have a leg along R. (See also (3.16) and the discussion preceding it.)Note that forms of the first type look like dτ ∧ φ, with φ ∈ Ω k−1 c (R × V ) a (k−1)-form with trivial contraction ι ∂τ φ = 0 along the vector field ∂ τ .Forms of the second type are ψ ∈ Ω k c (R × V ) satisfying ι ∂τ ψ = 0. Recall that the boundary condition only affects 0-forms and 1-forms, cf.(2.18).Concretely, we have that a 0-form ψ ∈ Ω 0 c (R × V ) satisfies the boundary condition if and only if its restriction to R × ∂V vanishes, a 1-form of the first type dτ ∧ φ ∈ Ω 1 c (R × V ) satisfies the boundary condition if and only if the restriction of φ ∈ Ω 0 c (R × V ) to R × ∂V vanishes, and every 1-form of the second type ψ ∈ Ω 1 c (R × V ) satisfies the boundary condition as a consequence of Lemma 3.5.
To prove the statement about the factorization, we have to show that, for every 1-form ) satisfying the boundary condition, the restriction of ( π ± ) * (ϕ) ∈ Ω 0 c (V ) to the boundary ∂V vanishes.For 1-forms of the second type this is trivial since ( π ± ) * (ψ) = 0 and for 1-forms of the first type dτ ∧ φ ∈ Ω 1 c (R × V ) this follows from the fact that φ ∈ Ω 0 c (R × V ) vanishes on R × ∂V as a consequence of the boundary condition.To prove that (5.6) is a quasi-isomorphism, we construct an explicit quasi-inverse by adapting the standard proof of the compactly supported Poincaré lemma, see e.g.[BT82,Proposition 4.6].Let us pick a compactly supported 1-form ω ∈ Ω 1 c (R) with unit integral R ω = 1 and define the cochain map (5.7) To simplify notation, we will always suppress pullbacks along the projection maps pr R and pr V on the two factors of R × V .The composition ( π ± ) * • ω * = id is the identity since the integral of ω ∈ Ω 1 c (R) is 1.The other composition is homotopic to the identity, i.e. id − ω * • ( π ± ) * = ∂K, for the cochain homotopy −1 defined by where τ −∞ denote restricted fiber integrations.It is easy to check that K is compatible with the boundary conditions by using their explicit description from the beginning of this proof.
Remark 5.4.The quasi-isomorphism ( π ± ) * in (5.6) describes how the linear observables F L ± ,c π −1 ± (V ) [1] of our 3-dimensional theory are identified with the linear observables Ω • ∂,c (V )[1] of a 2-dimensional theory.As some readers might prefer thinking in terms of fields instead of observables, let us dualize this construction using the evaluation pairings between linear observables and fields.The 3dimensional fields are described by the boundary conditioned Chern-Simons complex F L ± π −1 ± (V ) from (2.11) and the 2-dimensional fields by the boundary conditioned 1-shifted de Rham complex Ω • ∂ (V )[1], for the boundary condition demanding that the pullback of 0-forms to the boundary ∂V vanishes.(Note that there are no support restrictions in the field complexes.)The evaluation pairings are given by (2.14) in the 3-dimensional case and the analogous integration pairing in 2 dimensions.The dual of (5.6) is then given by the pullback map ( π ± ) * : Ω V ) on field complexes.Choosing any section of the principal R-bundle π ± : π −1 ± (V ) → V , we obtain an analogue of a Cauchy surface as per Remark 3.18 and can interpret this map as the constant evolution of initial data in Ω • ∂ (V )[1] along the fibers of π −1 ± (V ).Hence, the origin of the quasi-isomorphism (5.6) lies in the fact that Chern-Simons theory is locally constant (i.e.topological) along the fibers of π ± : π −1 ± (V ) → V .
Using Corollary A.4, we choose the orientation of B ± to be compatible with the orientation of M .With the induced orientation on V ⊆ B ± , the cochain complexes (5.5) can be endowed with the following unshifted linear Poisson structures which are natural with respect to the extension-by-zero maps ext V and we observe that fiber integration (5.6) defines a natural transformation (5.11) between functors from Open(B ± ) to PoCh R .Indeed, the component cochain maps ( π ± ) * are PoCh R -morphisms by Lemma 4.5.This allows us to obtain our desired weakly equivalent model for the AQFT ( π −1 ± ) * (A ± ) that is defined only in terms of geometric data on the 2-dimensional base manifold B ± .Explicitly, consider the functor 0 vanishes manifestly, see (5.9).
Proof.The statement follows directly from Lemma 5.3 and the fact that the CCR-functor preserves weak equivalences, see [BBS20, Proposition 5.3].
Non-dependence on the choice of chiral flow extension: In order to construct the linear Chern-Simons AQFT A ± via Green's homotopies, we have assumed extra data on M in the form of the proper R-action Θ ± : R × M → M .In the dimensionally reduced model B ± , this extra data is encoded as the quotient space B ± := M/ ∼ ± on which the AQFT is defined.
We now prove, under an additional topological assumption on M , that the dimensionally reduced theories B ± and B ± associated to two different choices of proper R-actions Θ ± and Θ ± on M are equivalent in an appropriate sense.Since A ± is fully determined by its dimensionally reduced counterpart, it follows that our construction is independent of the particular choice of proper R-action Θ ± on M .The important content of Assumption 3.23 for our construction is therefore only: (a) a choice of chirality ± on ∂M , and (b) the existence of a proper R-action on M which restricts on ∂M to a chiral flow of the chosen chirality.
Recall that the quotient space B ± of M by the proper R-action Θ ± is a smooth 2-manifold with orientation suitably compatible with the orientation of M , see Corollaries A.2 and A.4. Proposition 5.6.Suppose that Θ ± and Θ ± are two proper R-actions on the oriented 3-manifold M which restrict to the boundary ∂M as per (3.25) to give chiral flows Θ ± and Θ ± of the same chirality.If each component of M has more than one distinct topological end, then there exists an orientation-preserving diffeomorphism f : B ± → B ± between the quotient spaces of M by Θ ± and Θ ± .
Proof.We describe the case where M is connected, hence B ± and B ± are also connected.The result extends component-wise for disconnected M .Since any principal R-bundle is trivializable, there exist diffeomorphisms M ∼ = R × B ± ∼ = R × B ± .The hypothesis that M has more than one end implies that that B ± is compact.Indeed, if B ± were non-compact then the product M ∼ = R × B ± of connected, locally connected, non-compact spaces would have exactly one end, see [ILM14, Theorem 5.2].Similarly, B ± is also compact.
Since R deformation retracts to a point, it follows that B ± and B ± are homotopy equivalent, and hence have the same genus.Since the boundary restrictions Θ ± and Θ ± of the R-actions have the same chirality, it follows that ∂B ± = ∂M/ ∼ ± = ∂M/ ∼ ± = ∂B ± , see Remark 3.10.Then by the classification of orientable compact surfaces (see Remark 5.7 below) there exists an orientation-preserving diffeomorphism f : B ± → B ± .
Remark 5.7.The preceding proof exploits the classification of orientable compact surfaces.This may be stated as follows [Hir76, Chapter 9, Theorem 3.7]: For a connected, compact, orientable surface B of genus g with b boundary components, there exists a diffeomorphism B → Σ g,b , where Σ g,b is the g-holed torus with b disjoint open discs removed.When B is also oriented, we may upgrade this classification to use diffeomorphisms that preserve orientations.This is the form of the classification theorem used in the proof of Proposition 5.6.There always exists an orientation-reversing self-diffeomorphism r : Σ g,b → Σ g,b .It is a trivial adaptation of the b = 0 case [Mül09, Theorem A] to construct one.Specifically, Σ g,b may be embedded in R 3 reflection-symmetrically about a plane, say z = 0 in Cartesian coordinates.Then reflection about the plane (x, y, z) → (x, y, −z) gives r.If the diffeomorphism B → Σ g,b established above does not preserve orientations, we post-compose it with r to obtain a new diffeomorphism which does.
Remark 5.8.In Proposition 5.6, the hypothesis on topological ends of M is precisely an R-actionagnostic way to impose that the quotient spaces B ± and B ± are compact. 4This compactness allows us to address the cancellation problem (i.e.show that R × B ± ∼ = R × B ± implies B ± ∼ = B ± ) using the classical classification of compact surfaces.There exist analogous classification results for non-compact surfaces [BM79] but their application here would require a detailed analysis of the ends of B ± and B ± , which we avoid for simplicity.
Many spaces M of interest satisfy the hypotheses in Proposition 5.6, e.g. the filled cylinder R×D 2 (Example 3.27) and the annular cylinder R×A 2 where A 2 := {(x, y) ∈ R 2 | 1 2 ≤ x 2 +y 2 ≤ 1 } is an annulus.We note, however, that some important examples do not, including the half-space (Example 3.26).To complete the treatment of such an example, one should either generalize 4 Compactness of the quotient space B± implies that each component of M ∼ = R × B± has two distinct ends.To see this in case of connected M , take the two decreasing sequences Proposition 5.6 appropriately, or prove an analogous statement specific to the example of interest using ad hoc methods.
Suppose that f : B ± → B ± is an orientation-preserving diffeomorphism.Recall that the Poisson structure σ (0) of (5.9) is defined using only wedge products and integration of forms, and so it is compatible with pullbacks along orientation-preserving diffeomorphisms.Hence, pulling forms back along f defines a natural isomorphism (5.14) between functors from Open(B ± ) to PoCh R .By abuse of notation, we denote the functor f : Open(B ± ) → Open(B ± ) by the same symbol as the diffeomorphism which induces it.
Corollary 5.9.Suppose that the hypothesis of Proposition 5.6 holds and denote by f : B ± → B ± the resulting orientation-preserving diffeomorphism.There is a natural isomorphism between AQFTs on Open(B ± ), defined by applying the CCR-functor to the natural isomorphism (5.14).

Analyzing the dimensionally reduced AQFT
The aim of this section is to analyze some of the physics encoded in the dimensionally reduced AQFT B ± : Open(B ± ) → dg * Alg C from Corollary 5.5.In particular, we shall show that, restricting this AQFT to the interior IntB ± , one recovers the usual linear Chern-Simons theory on surfaces without boundary and, restricting to a tubular neighborhood of the boundary ∂B ± , one finds a chiral free boson.

Restriction to the interior
given by canonical quantization of the 1-shifted compactly supported de Rham complex along the linear Poisson structure (5.9).Observe that the latter is (a homological refinement of) the Atiyah-Bott Poisson structure [AB83] for flat connections on surfaces, hence the restricted AQFT B ± | IntB ± can be interpreted in terms of linear Chern-Simons theory on IntB ± .A truncation of this AQFT was previously studied in [DMS17].
Restriction near the boundary ∂B ± : In order to study the induced theory on the boundary ∂B ± , we consider a smooth embedding ∂B ± × [0, 1) → B ± as a tubular neighborhood of the boundary ∂B ± , so its restriction to ∂B ± × {0} embeds as ∂B ± .
Let us denote by Open(∂B ± ) the category whose objects are all non-empty open subsets W ⊆ ∂B ± of the 1-dimensional boundary manifold ∂B ± and whose morphisms ι W W : W → W are subset inclusions W ⊆ W ⊆ ∂B ± .We endow this category with the disjointness orthogonality relation, i.e. (W 1 ⊆ W ) ⊥ (W 2 ⊆ W ) if and only if W 1 ∩ W 2 = ∅, and introduce the orthogonal functor Open(∂B ± ) → Open(B ± ) defined by W → W × [0, 1).We denote by B ± | ∂B ± : Open(∂B ± ) → dg * Alg C the restriction of the AQFT B ± from Corollary 5.5 along this orthogonal functor.Explicitly, this theory assigns to (W ⊆ ∂B ± ) ∈ Open(∂B ± ) the dg-algebra We will now construct a weakly equivalent model for the AQFT B ± | ∂B ± that is determined only from 1-dimensional data on ∂B ± .
For the following construction, we choose a global coordinate r on [0, 1) and pick any compactly supported 1-form ω ∈ Ω 1 c [0, 1) with unit integral 1 0 ω = 1.As before, we simply write ω := pr * [0,1) (ω) ∈ Ω 1 W × [0, 1) for the form obtained via pullback along the projection map, and similar for pullbacks of forms from W .We define, for each non-empty open subset (W ⊆ ∂B ± ) ∈ Open(∂B ± ), the map Regarding the domain of this map as a cochain complex concentrated in degree 0 (with trivial differential), one easily checks that κ defines a cochain map.Indeed, for all ϕ ∈ Ω 0 c (W ).
Proof.The proof is similar to the one of [GRW22, Theorem 4.1], so we can be relatively brief.A quasi-inverse for κ is given by the following fiber integration which is a cochain map owing to the boundary condition on Ω 0 ∂,c W × [0, 1) .One checks that λ • κ = id since ω integrates to 1.The other composition is homotopic to the identity, i. Remark 6.2.Building on Remark 5.4, let us also describe the quasi-isomorphism κ in (6.3) using the dual language of fields instead of observables.It turns out to be simpler to interpret the quasi-inverse cochain map λ in (6.5).The 2-dimensional fields are described by the boundary conditioned 1-shifted de Rham complex Ω • ∂ W × [0, 1) [1] and the 1-dimensional fields by the complex Ω 1 (W ) that is concentrated in degree 0. (One should think of elements J ∈ Ω 1 (W ) as the Abelian currents of a chiral free boson.)The dual of λ is then given by the pullback map pr * W : Ω 1 (W ) → Ω • ∂ W × [0, 1) [1] along the projection pr W : W × [0, 1) → W . Thinking of W as the boundary W × {0} of W × [0, 1), one can interpret this map as the constant evolution of boundary values in Ω 1 (W ) along the [0, 1)-factor.Hence, the origin of the quasi-isomorphism (6.5), and hence of (6.3), lies in the fact that Chern-Simons theory on W × [0, 1) is locally constant (i.e.topological) normal to the boundary.of AQFTs on Open(∂B ± ), defined by applying the CCR-functor to the natural transformation (6.8), is a natural weak equivalence.Hence, the AQFT C ± from (6.9), which describes the chiral free boson, provides a weakly equivalent model for the boundary restriction B ± | ∂B ± of the dimensionally reduced bulk/boundary AQFT from Corollary 5.5.
Proof.This follows directly from Lemma 6.1 and the fact that the CCR-functor preserves weak equivalences, see [BBS20,Proposition 5.3].

Interplay between bulk and boundary:
The AQFT B ± : Open(B ± ) → dg * Alg C from Corollary 5.5 encodes also the interplay between Chern-Simons observables in the bulk and chiral free boson observables on the boundary.For instance, given any open subset of the form W × [0, 1) ⊆ B ± in the tubular neighborhood of the boundary, we obtain a morphism ι : W × 1 2 , 1 −→ W × [0, 1) (6.11) in the category Open(B ± ) whose source is an interior region in the bulk and whose target is a region that intersects the boundary.(The particular choice of the open interval ( 1 2 , 1) ⊆ [0, 1) is inessential and purely conventional.Since Chern-Simons theory is a topological QFT in the bulk, we could choose equally well another open interval (a, b) ∈ [0, 1).)Applying the functor B ± : Open(B ± ) → dg * Alg C on this morphism and using the natural weak equivalence from Corollary 6.3 yields the following zig-zag o o (6.12) Lemma A.3.The canonical orientation on R induces a canonical bundle orientation on any principal R-bundle π : N → B.
Proof.Recall that a bundle orientation on π : N → B can be expressed as an orientation of the vector bundle V N ⊆ T N of vertical vectors with respect to π, see e.g.[GHV72, Section 7.4, Proposition II].For any element a ∈ R in the Lie algebra of the structure group R, there is a fundamental vector field A ∈ Γ ∞ (V N ) on the total space N of the principal bundle, see e.g.[CDD82, Section A.1].These fundamental vector fields give R-linear isomorphisms R → V p N , a → A p between the Lie algebra and the space of vertical vectors at any p ∈ N .
The canonical orientation on the Lie group R identifies bases {a} of its Lie algebra R as positively oriented if a > 0. The bundle orientation on π : N → B is then induced by transporting along the isomorphisms above, i.e. {A p } is positively oriented in V p N for a > 0. This is a smooth choice of positively oriented bases since the fundamental vector fields A are smooth.Proof.Consider first the case where either set of orientations is empty, i.e. either N or B is non-orientable.Since any principal R-bundle is trivializable N ∼ = R × B and R is orientable, N is orientable if and only B is orientable.Hence, in this case both sets of orientations are empty and one has the claimed bijection.
Consider now the case where both N and B are orientable.It is trivial to produce a compatible orientation on N from any orientation on B, using the definition of compatibility.For an inverse, when given an orientation on N , choose an arbitrary orientation on B and transform it to a compatible one by reversing it on those components where it is incompatible.

A.2 Properties of Θ-future/past sets
The Θ-future/past sets entering property (ii) of Definition 3.12 may be expressed in terms of a binary relation ≤ Θ on N associated to the R-action Θ : R × N → N , defined as p ≤ Θ q if ∃s ≥ 0 such that q = Θ(s, p) . (A.1) The Θ-future/past J ↑/↓ Θ (S) of a subset S ⊆ N are respectively the successor-and predecessor-sets of S with respect to ≤ Θ .Compare this to Remark 3.6.

Example 3. 3 .
The terminology in Definition 3.2 is motivated by the graphical representation of such tangent vectors in the standard Minkowski spacetime (see Example 2.1): a quick calculation one checks that the tangent vector ∂ + := ∂ ∂x + is -pointing and the tangent vector ∂ − := ∂ ∂x − is -pointing.The additive inverses −∂ + and −∂ − are, respectively, -pointing and -pointing.

Lemma A. 5 .
For any proper smooth R-action Θ : R × N → N on a smooth manifold N , the (graph of the) relation ≤ Θ defined in (A.1) is a closed subset of N × N .Proof.The (graph of the) relation ≤ Θ ⊆ N × N of (A.1) may be expressed as the image≤ Θ = θ ([0, ∞) × N ) ⊆ N × N (A.2)of a closed set under the shear map θ : R × N → N × N associated to R-action Θ as per (3.9).By hypothesis, Θ is a proper R-action i.e. θ is a proper map.Since N × N is a manifold, so Hausdorff and locally compact, it follows by [Lee13, Theorem A.57] that θ is a closed map.Hence the relation ≤ Θ is a closed subset in N × N .
Let us further endow the boundary ∂M = R 2 with the standard Minkowski metric g = −dt 2 + dx 2 and the time-orientation t = [∂ t ].The induced orientation of the boundary is o ∂M = [dt ∧ dx].Taking the standard basis (dt, dx) of Ω 1 (∂M ), one has that * dt = −dx and * dx = −dt.