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Fully extended BV-BFV description of General Relativity in three dimensions

We compute the extension of the BV theory for three-dimensional General Relativity to all higher-codimension strata - boundaries, corners and vertices - in the BV-BFV framework. Moreover, we show that such extension is strongly equivalent to (nondegenerate) BF theory at all codimensions.


Introduction
The BV-BFV formalism is a combination of a Lagrangian approach to field theories with on-shell symmetries -the BV formalism -and of its counterpart for constrained Hamiltonian systems -the BFV formalism -named after the work of Batalin, Fradkin and Vilkovisky [BV77; BV81;BF83].
The link between the two approaches was developed by Cattaneo, Mnev and Reshetikhin in [CMR14] as a first step towards quantisation of gauge theories on manifolds with boundaries, with an axiomatisation of Classical and Quantum field theory in mind. The main idea is to allow boundaries to spoil gauge invariance of a given theory, described by means of its BV-cohomology, but to control such failure by means of induced cohomological data associated to the boundary. This interplay between bulk and boundary data allows for a consistent quantisation scheme [CMR18], compatible with gluing by construction, whose output is the cohomology of a quantum operator that directly encodes gauge invariance.
The program of approaching General Relativity (GR) from this point of view was initiated in [Sch15], showing that when diffeomorphism symmetry is involved, the induction procedure essential for the BV-BFV correspondence to hold is far from being guaranteed. Indeed, while the standard example of General Relativity in the Einstein-Hilbert formulation satisfies all the axioms, thus providing a well defined BV-BFV theory in all spacetime dimensions d = 2 [CS16], obstructions arise in 4d General Relativity in the (Einstein-Sciama-Kibble-) Palatini-Cartan-Holst tetradic formulation 1 [CS17b]. Similar obstructions have been found in onedimensional reparametrisation models [CS17a], or are to be expected in certain supersymmetric models 2 [GP18].
In this paper we show that, in stark contrast with the four dimensional analogue, General Relativity in three spacetime dimensions, phrased in the triadic language [Wit89; Car98; Car22; Wis09], admits an extension to all higher codimension strata in the BV-BFV sense (Definition 3). This means not only that we can control the failure of gauge invariance of the theory upon introducing a boundary, but that the boundary gauge invariance is controlled by corner cohomology, and so on. One says in this case that 3d GR is a fully extended BV-BFV theory.
We obtain explicit expressions for the BV-BFV data at higher codimension strata, making 3d GR the first example of a fully extended theory that features a nontrivial symplectic reduction at every step, a procedure that fails in the analogous 4d theory. Out of such data, one can directly read relevant information such as the algebra of constraints together with a cohomological presentation of the reduced phase space (from the codimension-1 data), and the representations carried by boundary insertions (in codimension-2). Moreover, following [MSW19], a fully extended BV-BFV theory induces a solution of Witten descent equations [Wit88] -a step towards the understanding of observables in General Relativity -and one can discuss the emergence of edge modes and holographic counterparts (see, e.g. [CHv95]). Investigations in this direction will be carried out in a further paper.
Furthermore, by extending the notion of strong equivalence of BV theories and the result in [CSS18] to higher codimensions, we show that the fully extended BV-BFV description of GR is strongly equivalent to that of nondegenerate BF theory. This means that, on every stratum, the data of GR is equivalent to that of BF theory, with the action functionals being preserved by symplectomorphisms between spaces of fields 3 at every codimension.
One way of interpreting this result is by thinking of BF theory as a possibly degenerate version of GR, which coincides on an open sector, i.e. when the nondegeneracy condition is imposed. This requirement is not expected to spoil BV-BFV quantisation of BF theory (see [CMR18]).
Although it is well known that 3d General Relativity is classically equivalent to nondegenerate BF theory [Wit89], in this paper we explicitly write down the symmetries in terms of diffeomorphisms, and show that this description is equivalent to the one coming from standard symmetries of BF theory, at all codimensions. This 1 The physics literature for this version of GR seems to disagree on standard nomenclature. This is also why we refrain from naming a particular version of the theory in the title of the current paper.
2 It appears that models where a reduction of classical data is needed are more likely to encounter obstructions at the BV-BFV level. 3 One for every stratum ψ (k) : F (k) BF * . See Definition 3 to fix the notation.
is the result of nontrivial calculations, interesting mostly as they can be taken as a guideline or bootcamp for the more involved case of 4d gravity.
In Section 3 we describe in detail the constructive steps one needs in order to obtain the BV-BFV data at every codimension of a stratified manifold {M (k) } k=0...3 . We divide the proof of the main Theorem, stating that GR in the BV formalism is fully extended, into three Propositions, each of which is aimed at recovering data one codimension further.
In Section 4 we then show how explicit symplectomorphisms can be found between the spaces of fields F (k) GR/BF * at every codimension, and we show how they commute with the BV-BFV surjective submersion maps, essentially showing how reduction commutes with equivalence at codimension-k.
The results in this paper show how diffeomorphisms can be seen as an equivalent choice of a BV-extension of classical BF theory, and fully describe the compatibility with lower dimensional strata, completely characterising the symmetries of GR in three dimensions.

Preliminaries
The strategy employed in this paper is to consider the BV-data associated to a manifold M and, step by step, analyse what structure it induces if we allow M to carry a stratification of increasing codimension. We follow here the adaptation of the classical BV-BFV axioms introduced in [CMR14], as proposed by [MSW19].
Remark 2. A particular example of a stratification is given by a manifold with corners (and vertices, i.e. boundaries of corners), where the connected components of boundaries, corners and vertices compose the cells of a stratum M (k) .
Definition 3. A n-extended exact BV-BFV theory is the assignment, to a nstratification {M (k) } k=0...n (m ≥ n) of the data F ↑n = (F (k) , S (k) , α (k) , Q (k) , π (k) ) k=0...n such that for every k ≤ n: (1) F (k) is a graded manifold equipped with an exact symplectic form When n = m we say that the theory is fully extended. When n = 0, the data is that of a BV theory.

Remark 4. A direct interpretation of Equations
(1) is that, on every stratum, Q (k) is the Hamiltonian vector field of the action functional S (k) up to boundary terms (1a), and that the classical master equation is satisfied up to boundary terms (1b).
We will be concerned here with classical field theories that enjoy symmetries given by Lie algebra actions, and our starting point to build an extended (exact) BV-BFV theory is a couple (F cl , S cl ). The space of classical fields F cl is the space of sections 5 of some sheaf or bundle E → M , while S cl is a local functional on F cl , i.e. a function of the fields and a finite number of jets, called action functional. The symmetry data is sometimes encoded in an involutive distribution 6 D cl on F cl .
The first step is to build a BV theory (or 0-extended BV-BFV) F = (F, , S, Q), by promoting the space of classical field to a (−1)-symplectic graded manifold (F, ), and constructing a cohomological vector field Q on F, for which S is the Hamiltonian functional [BV77;BV81]. It is then possible to build an m-extended BV-BFV theory using a constructive approach.
In a practical scenario, this goes through by integrating by parts the terms in δS (n) , but the resulting data on the higher-codimension stratum does not automatically satisfy the axioms in Definition 3. In particular, the existence of the symplectic space F (n+1) (and hence of π (n) ) is not always guaranteed (see [CS17b;CS17a]). When this happens the theory is then only n-extendable. We summarise the previous discussion with the following definition: Definition 5. Let M be an m-dimensional manifold and let F ↑0 an exact BV theory on it. We say that the BV-theory F ↑0 is n-extendable if, for every n-stratification such that M (0) = M , there exist an n-extended exact BV-BFV theory F ↑n associated to it. If n = m we will say that F ↑0 is fully extendable.
Remark 6. In the literature, (1-extended) BV-BFV theories usually arise when considering a BV theory on a manifold with boundary. In Definition 3 this amounts to taking into account a filtration where M (0) = M is the manifold itself and M (1) = ∂M is the boundary of the manifold. The generalization of this to higher codimensions is to consider k-extended BV-BFV theories on manifolds with boundary, corners, vertices, etc. and consider the filtration given by M (k) .
We give here the notion of a strong 7 equivalence of (extended) BV-BFV theories. It implies the standard notion of equivalence of classical field theories, which requires the critical loci of two action functionals to be isomorphic (modulo symmetries). In the case of BV theories (i.e. the 0-extended version of this) the following definition has been proposed in [CSS18]: 5 In the present paper we will consider principal connections as fields. However, we can reduce to this setting by expanding around an arbitrary reference connection. 6 In full generality the BV formalism only requires that D cl be involutive on the critical locus of S cl , i.e. the space of solutions of the associated variational problem. 7 The natural notion of equivalence, given the cohomological context in which physical data is presented, would coincide with weak equivalences of BV-BFV complexes (see [MSW19]). The definition we propose here is essentially that of an isomorphism of complexes, hence a stronger requirement.
Definition 7. A strong equivalence between two BV theories F ↑0 1 and F ↑0 2 is a degree-0 symplectomorphism 2 ) preserving the BV action 8 : We can modify this definition to encompass n-extended BV-BFV theories.
Definition 8. A strong equivalence between two n-extended exact BV-BFV theories F ↑n 1 and F ↑n 2 is a collection of degree-0 symplectomorphisms 2 ) preserving the k th BFV action: 1 .

Setup
The common framework shared by 3 dimensional General Relativity and BF theory is as follows. Let P → M be an SO(2, 1)-principal bundle on a 3-dimensional, compact, orientable 9 manifold M . Let also V be the associated vector bundle where each fibre is isomorphic to (V, η), a 3-dimensional vector space with the Minkowski inner product η on it. We further identify so(2, 1) ∼ = 2 V using η and we define a map Tr: 3 V → R given by the volume form and such that Tr To keep the notation light we will use the shorthand In the following subsections we specify the details proper to each theory.
2.1. Three-dimensional BF theory. The fields of the theory are B ∈ Ω 1 (M, V) and a connection A ∈ A P . We will think of A as a connection form around the trivial connection, that is to say A ∈ Ω 1 (M, 2 V).
Definition 9. Classical BF theory is the pair (F cl BF , S cl BF ) where is the space of fields, and the action functional reads with F A ∈ Ω 2 (M, 2 V) the curvature of the connection A. We can further require B to be nondegenerate as a map B : T M → V η 2 V. Denoting by Ω 1 nd (M, 2 V) the space of nondegenerate B's, we will call the resulting theory nondegenerate BF theory, and denote it with the notation BF * where relevant. 8 We always consider action functionals S (0) modulo constants. 9 Extensions to noncompact manifolds are possible, but outside the main objective of this paper. One possible way out is to define fields as compactly supported sections of the bundles on the noncompact components of a manifold. Orientability is not necessary, but we restrict to orientable manifolds for simplicity.
The symmetries of the theory comprise gauge trasformations, parametrized by 10 χ ∈ Ω 0 (M, so(2, 1)) Ω 0 (M, 2 V) and by what is sometimes referred to as shift symmetry, a traslation of B parametrized by τ ∈ Ω 0 (M, V), We recall the BV version of three-dimensional BF theory.
Definition 10. The BV-data for BF theory is given by F ↑0 BF = (F BF , α BF , S BF , Q BF ), where the BV space of fields can be written as and, if we arrange the fields in the following convenient way the BV data reads 11 If B ∈ Ω 1 nd (M, 2 V) we will denote the resulting BV theory by F ↑0 BF * . As every AKSZ theory [Ale+97], BF theory can be fully extended: Theorem 11. [CMR14] The BV theory F ↑0 BF = (F BF , S BF , α BF , Q BF ) is fully extendable. The BV-BFV data of the fully extended theory F ↑3 BF is given by the following expressions (i = 0 . . . 3): where we used once again the convention that only the admissible terms appear in the integrands and π (i) BF is the restriction of the fields to M (i) . Note that in this notation BF theory is self-similar, i.e. the action S, the symplectic two form ω and the cohomological vector field Q have the same expression on bulk (0-stratum), boundary (1-stratum) and every subsequent iteration. 10 We will denote here the action of a symmetry by the notation δχ with parameter χ. This is a notation historically used to denote a χ-dependent vector field acting on generators on the algebra of functions over F cl . It will be replaced by a well-defined vector field when we pass to the BV formalism. 11 We use here the convention that only the admissible terms (i.e. the ones that are top forms) appear in the integrands.

2.2.
Three dimensional General Relativity. The fields are a co-frame field e ∈ Ω 1 nd (M, V), also called a triad (nd stands for non degenerate), i.e. an isomorphism e : T M → V, and an SO(3, 1) principal connection ω ∈ A P Ω 1 (M, 2 V) (again we work around the trivial connection).
Definition 12. Classical three dimensional General Relativity (GR) is the pair is the space of fields and the action functional reads In order to define a BV theory extending (12), we have to incorporate the symmetries by extending the space of fields. The classical functional S cl GR is invariant under the action of internal gauge transformations SO(2, 1) and the action of spacetime diffeomorphisms. We parametrize them with two ghost fields, c ∈ Ω 0 [1](M, ∧ 2 V) and locally ξ ∈ Γ[1](T M ) respectively: is the graded commutator between the contraction with respect to ξ (a degree-0 derivation), and d ω is the covariant derivative (a degree-1 derivation). With these quantities we can also define ι [ξ,ξ] The BV structure associated to these symmetries has been studied in generality in [CS17b, Section 3].
Definition 13. The BV theory for General Relativity in three dimensions is given by the data F ↑0 T M , the BV one-form and action functional are and with the vector field Q GR given by the defining property ι Q GR GR = δS GR , when M is closed and without boundary. Its explicit expression is readily found to be (dropping the GR-subscript): where the bullet denotes the 1-form coefficient of elements in We can now state the main results in this paper. Sections 3 and 4 will be devoted to their proof.
The fully extended BV-BFV theories F ↑3 GR and F ↑3 BF * are strongly equivalent.
Remark 16. The space of BV fields for classical GR and the action of symmetries presented in Definition 13 is essentially independent of spacetime dimensions (although the action functional is not). However, it was shown in [CS17b] that the 4-dimensional BV theory of General Relativity (in the tetrad formalism) cannot be extended without extra assumptions on the fields. Theorem 14, compared with the no-go result in [CS17b] marks a stark difference between 3 and 4 spacetime dimensions. GR in the Einstein-Hilbert formalism, instead, is independent of spacetime dimensions and is always at least 1-extendable (outside of d = 2) [CS16].

Fully extended BV-BFV structure of GR
In this section we will prove Theorem 14 by extending the BV theory of definition 13 step by step and build the fully extended BV-BFV data for GR at every codimension.
First, let us introduce some useful notation and explain the common strategy to prove each step. Throughout the section, as we will only consider GR theory, we will drop the GR subscript everywhere, except in stating the results.
3.1. Notation and strategy. Consider X ∈ Ω 1 (M, V) and Y ∈ Ω 1 (M, 2 V). Since the image of a nondegenerate triad e is a basis of V at every point, we can express X and Y as where e i = e(∂ i ), and we denote by X (i) the i-th component of X and by Y (ij) the ij-th component of Y . We can then define the following projections, i = 1, 2, 3: It is also useful to define a dual map acting on elements of Ω 1 (M, 2 V) : We outline here the strategy of the proofs of the construction of the maps π (i) of Definition 3, as was introduced in [CS16]. We analyse here the extension to a codimension-1 stratum, and the procedure can be reproduced in a straightforward way to obtain the reduction to other codimensions.
The goal is to construct data corresponding to the codimension-1 stratum (the 1-extended theory). We will assume here for simplicity that M (1) = ∂M is a boundary. The first step is to consider the variation of the action functional in the bulk and to construct appropriate data to satisfy (1a) and the axioms in Definition 3. The variation of δS will consist of two terms generated by integration by parts: the bulk term will be interpreted as ι Q -defining the Euler-Lagrange equations for the variational problem -while the remainder, a boundary term, is interpreted as a one-formα on some appropriate space (see below). Namely, we have an equation formally equivalent to (1a) δS = ι Q −π * α , but such that the correction termπ * α lives on the intermediate spaceF, defined as the space of fields and transversal jets restricted to the boundary, withπ : F →F being the restriction of fields to ∂M . We will callF space of pre-boundary fields.
We defineˇ := δα, which is a closed form, but in general degenerate and hence not symplectic. However, one expects it to be pre-symplectic i.e. such that the kernel of the associated map is regular. This condition, which has to be explicitly checked, shows that the kernel is a subbundle in TF, and one can define the space of boundary fields to be the symplectic reduction 12 : The symplectic reduction map π red :F −→ F (1) can be computed in a chart by explicitly flowing along the vertical vector fields (i.e. X ∈ TF such thatˇ (X) = 0). The projection π (1) : F → F (1) to the true space of boundary fields, is then obtained by composingπ with the symplectic reduction map π red : π (1) := π red •π. See the proof of Proposition 17 for more details.
To recover the rest of the BV-BFV data on the higher codimension stratum (the boundary), we first define a pre-boundary action functionalŠ onF via an analogue of Equation (1b): ι Q ι Q = 2π * Š . It then follows from the BV-BFV theorems [CMR14] thatŠ is basic with respect to π red :F → F (1) : namely, there exists S (1) on F (1) such that π * red S (1) =Š, and consequently ι Q ι Q = 2π (1) * Š . This procedure can be repeated at every codimension, the integration by parts connecting the variation of the codimension-k action functional with the codimension-(k + 1) one-formα.
3.2. 1-extended GR theory. If we allow M to bear a 1-stratification, and denote codimension-1 strata by M (1) , denoting by Ω 1 nd (M (1) , V) the space of maps e : T M (1) −→ V such that the image of e is a linearly independent system, we have the following.
GR . The codimension-1 data are: • The space of codimension-1 fields, given by the bundle with local trivialisation on an open together with a fixed vector field n ∈ Γ(V), completing the image of elements e ∈ U (1) to a basis of V; • The projection to codimension-1 fields • The codimension-1 one-form, symplectic form and action functional (1) where X n = L ω ξ ( n ξ n ) and Y n = [ c, n ξ n ], and the superscript (n) denoting the component with respect to n .
It is then possible to solve equations (11) to yield and, since Equations (13) force the left hand sides to be proportional to the odd field ξ n , Equations (12) are automatically satisfied. This shows that the kernel has constant rank -ˇ is pre-symplectic -and we can perform symplectic reduction. We now compute the BV-BFV data by presenting a chart for the symplectic reduction We flow along vertical vector fields (i.e. vector fields in the kernel ofˇ ) to obtain boundary coordinates. In other words, denoting by ϕ Y the flow of a vector field Y at time s = 1, we define the change of coordinates on a field ψ to be given by where the vertical vector fields E n , E † n , C n , Ω † n , X n and X a read E n = (X en ) δ δe n − (X en ) (a) e † a ξ n + (X en ) (n) e † n ξ n δ δω − ι ξ (X en ) (a) e † a ξ n + (X en )e † n ξ n δ δc Using X a and X n we can eliminate χ a and χ n . From the differential equatioṅ χ ρ (s) = (X χρ ) (ρ = a, n), we conclude that χ ρ (s = 1) = 0 ⇐⇒ (X χρ ) = −χ ρ (0), with s the parameter on the diffeomorphism integrating the vector field in an interval. It follows that where the numbered superscript [1] denotes the step-by-step reconstruction of the change of variables: for instance, the variable p † n e † gets transformed along the flow of X n , as well as Ω † n and E † n . Flowing along each of these vertical vector fields defines a temporary change of coordinates, which will be denoted by the superscript [k] .
Indeed, using Ω † n we have that ω † n (s = 1) = 0 ⇐⇒ (X ω † n ) = −ω † n which induces: [1] + ω † na ξ n e † (a) (16) Moving on to E † n we solve the associated differential equations to yield − (e † n ι ξ c † n ξ n ) (n) (17) while using C † n in the same fashion we can conclude: Notice that we did not consider the coefficient of δ δe † in E † n , for have fixed the values c † n = ω † n = 0 at the internal parameter s = 1 along the flow of the previously employed vector fields. Now it is time to turn to E n . Its simplified expression after flowing along the other vector fields is We want to flow along its integrating diffeomorphism and set the field e n to a given value. In this case we cannot set e n (s = 1) = 0 because this would violate the nondegeneracy requirement for the tetrad field. We will fix it to a vector n ∈ V proportional to the original e n , pointwise, and thus linearly independent from (the vectors in the image of) e. Observe that for an open subset U (1) ⊂ Ω 1 nd (M (1) , V) the choice of n is independent of e ∈ U (1) . The differential equationė n = (X en ) is solved as e n (s) = e n (0) + (X en )s, so that, fixing e n (s = 1) ≡ (1 + ε)e n (0) yields the flow: (X en ) = εe n (0); e n (s) = e n (0)(1 + s) In order to compute the other flows, we have to consider the components of (X en ) in the (varying) basis vectors {e a }. With our choice we have (X en ) ∝ e n (0) ∝ e n (s).
This data defines a chart for the (locally trivialised) bundle since, for all e ∈ U we can fix a completion n that does not depend on e. An easy computation shows thatα is not basic (in particular it is not horizontal, i.e. ι Enα = 0), but it descends to the quotient upon adding the term δ(ee † n ξ n ). In the local chart defined by (22), we define: 14 α (1) := Tr M (1) − eδ ω + ω † δ c − p n e † n δ ξ n − ι δ ξ e p n e † − ι δ ξ e p a e † + ι ξ ω † δ ω 13 As mentioned in Section 3.1, most of the times we will omit mentioning precomposition witȟ π and identify the BV-BFV reduction map with symplectic reduction. 14 Since δ n = 0 we obtain, in the space of preboundary fields, δεen = −(1 + ε)δen. Hence nδ ξ n = (1 + ε)enδ(1 + ε) −1 ξ n + enδξ n = −(1 + ε)en(1 + ε) −2 δεξ n + enδξ n = δenξ n + enδξ n . so thatα + δ(ee † n ξ n ) = π * red α (1) and, consequently, (1) := Tr Observe that there are now two possible bases in V, one given by the restriction of e to the boundary {e n , e a } and the transformed one { n , e a } (these vectors still form a basis since the additional terms are nilpotent). We then have two possible sets of projections p † , p † defined as in Section 3.1. We will denote with a tilde the one referring to the transformed boundary basis { n ,ẽ a }. Using the fact that n is parallel to e n by construction, there is a natural diffeomorphism ( p a e † , p n e † ) → e † with e † = p a e † + p n e † . Let now p † n the projection, as described above, with respect to n . Then the inverse is given by e † → ( e † − p † n e † , p † n e † ). We extend this diffeomorphism to the whole space of fields with the identity. Hence the boundary forms can be more conveniently rewritten as Finally, we can computeŠ such that ι Q ι Q = 2π * Š , using Equation (4) for the bulk Q. A straightforward calculation yields: ξ] ee † + e n ξ n d ω ι ξ e † + ι ξ ed ω (e † n ξ n ) + e † n ξ n ι ξ d ω e + e n ξ n d ω (e † n ξ n ) − 1 2 ι [ξ,ξ] χVξ n − ι ξ dξ n χ n Vξ n .
Using the projection (8) we can find the codimension-1 action functional to be . As a direct consequence, then, we have Having found the codimension-1 action functional we can compute the cohomological vector field Q (1) on the 1-stratum. Since the coordinates we are using are not a Darboux chart, some complications in the computation arise. Nonetheless, the non-degeneracy of (1) guarantees that starting from the variation of the action and using the equation ι Q (1) (1) = δS (1) , we can compute the cohomological codimension-1 vector field Q (1) .
Remark 18. Observe that, strictly speaking, Equation (1a) is satisfied by the above data only if we modify S GR by the boundary term M (0) d(ee † n ξ n ), so that the associated pre-boundary one-formα is automatically basic. Indeed, this is necessary only if we insist on the BV-BFV data to be exact, i.e. such that the symplectic forms are exact at every codimension (k) = δα (k) and that their symplectic potential α (k) is pulled back to the respective term in (1a). A picture suitable to situations like the present one, where symplectic reduction is possible, but nontrivial, at every codimension, is to consider the symplectic forms instead of their potentials, and the equation which follows from (1a) by differentiating w.r.t. δ. However, modifying S (k) by a term concentrated in codimension-(k + 1) does not change the BV-BFV structure (cf. [MSW19]).
Remark 19. The expression (9b) is not in the Darboux form. The change of coordinates to a Darboux chart will turn out to coincide with the boundary symplectomorphism between GR and BF theory (see Section 4.2). The same symplectomorphism will also turn the local trivialisation (7) into a global one, showing that the bundle F (1) We are now ready to compute the structure induced on codimension-2 strata when M carries a 2-stratification, for example in the presence of corners. Building up from the codimension-1 BV-BFV structure found in Proposition 17, denoting again by Ω 1 nd (M (2) , V) the space of maps e : T M (2) −→ V whose image defines a linearly independent system, we have the following result.
The one-formα (1) is not basic w.r.t. π red , and it descends to the quotient only upon adding the exact term δ(ι ξ e e † m ξ m + n ξ n e † m ξ m ). The codimension-2 one-form is then given by cδ e + ι ξ eδ ω + m ξ m δ ω + n ξ n δ ω.
Proposition 21. The BV theory F ↑0 GR is 3-extendable to F ↑3 GR . The codimension-3 data are: • The space of fields (3) • The cohomological vector field Q Remark 22. Since M (3) is a set of points, integration is here intended as sum over such points, with the space of fields on a single point being given by Proof. In order to keep the notation light and readable we drop the tildes. Once again, from the defining equation ι Q (2) (2) = δS (2) +π (2)α(2) we geť n δξ n δc + m δξ m δc − δe a ξ a δc + e a δξ a δc, withα (2) ∈ Ω 1 (F (2) ), the space of restrictions of fields in F (2) (and their normal jets) to the stratum M (3) , withπ : F (2) −→F (2) . There is only one nontrivial equation defining the kernel of (ˇ (2) ) : n X ξ n + m X ξ m + e a X ξ a − X ea ξ a = 0. This is solved as in the previous cases by X µ ξ = (X ea ) (µ) ξ a and, since this defines a subbundle of TF (2) , it is possible to perform the symplectic reduction. The only vertical vector field is Once again, flowing along E a we are able to fix the vector e a to a = (1 + )e a for some constant and consequently, ξ a[1] = (1 + )ξ a , while the rest is left unchanged. This defines the symplectic reduction π red :F (2) −→ F (3) , and the BV-BFV map π (3) * := π red •π. Then, the expression . Lastly, we compute the vertex action. With a calculation completely analogous to what was done in Propositions 17 and 20 we computě S (2) such that ι Q (2) ι Q (2) (2) = 2π * Š(2) : Then, we get that the vertex action [c, c] n ξ n satisfiesŠ (2) = π * red S (3) , and consequently ι Q (2) ι Q (2) (2) = 2π (3) * S (3) .

BV-BFV equivalence
The goal of this section is to prove the following theorem based on Definition 8.
Theorem 23. The fully extended BV-BFV theories F ↑3 GR and F ↑3 BF * are strongly equivalent.
Remark 24. Explicitly, we have to prove the existence of invertible vertical arrows that make the following diagram commute.
Note that the vertical symplectomorphisms preserve the action functionals, i.e. they satisfy (ψ (k) ) * S (k) GR , and that the horizontal arrows on both lines have been already described in sections 2.1 and 2.2 respectively.
Once more we recall that one can think of an example of stratification {M (k) } as given by the data of a the three-dimensional manifold M ≡ M (0) with vertices, i.e. a manifold whose boundary has boundaries of its own, called corners, which also have boundaries, called vertices.
4.1. Equivalence on the bulk. A strong equivalence (see Definition 7) between the BV data associated to BF theory and GR was proven in [CSS18, Theorem 10], provided that on B is imposed a non degeneracy condition. We denote by BF * nondegenerate BF theory (see Definition 9). An explicit generating function for the canonical transformation between the two (−1)-symplectic spaces of fields has been given as well. We recall here the most important steps of the construction.
Lemma 25. The symplectomorphism ψ : F GR → F BF * is given by ψ : whereas its inverse is given by the following formula ψ −1 : where the indices (a) denote components with respect to the basis {B a }.
Proof. We first apply (42) to the generating function (41) and get Since this is an equation between objects valued in ∧ 3 V , we can extract an e a factor and get Having τ † , we can now invert all other equations. An easy but lengthy computation shows that this symplectomorphism correctly satisfies GR = ψ * BF * and it preserves the action functionals, as was shown in [CSS18].
Remark 26. In order to build the inverse symplectomorphism ψ −1 we must require that the image of B be a basis of V at every point. Thus the two theories are strongly equivalent only if also B satisfies the non-degeneracy condition in Definition 9.
4.2. Equivalence on boundaries, corner, vertices. Since the form (1) GR of Equation (9b) is not in the Darboux form, it is not possible to find a generating function. We can nonetheless produce an explicit symplectomorphism and its inverse: (46) where superscripts (a) denote the components with respect to { n , B a }. It is straightforward to check that ψ (1) • (ψ (1) ) −1 = id, (ψ (1) ) −1 • ψ (1) = id and (ψ (2) ) −1 : while, on the vertex, we have with inverses given by ξ m = τ (m) , ξ n = τ (n) and ξ a = τ (a) , i.e. the components of τ with respect to m , n and a respectively. Finally, it is straightforward to check that (ψ (k) ) * S (k) BF * = S (k) GR for k = 1, 2, 3. 4.3. Commutativity. In this section we prove the commutativity of the three square subdiagrams of the diagram (40). This is sufficent to prove commutativity as a whole. For the sake of clarity, we denote the BF variables on the 1-stratum (and subsequent 2-and 3-strata) with a tilde, analogously to the GR notation. To avoid confusion, we explicitly denote the restriction to the 1-stratum (resp. 2-and 3-stratum) with an apex ', e.g. e ≡ e| M (1) and e ≡ e| M (2) .

Cosmological constant
In this section we consider BF theory and GR theory with the addition of what is generally known as the cosmological term: a cubic term in B (respectively e). Classically this amounts to considering the functionals Since the additional cosmological term does not contain any derivative, also GR is fully extendable and the reductions are not modified. The actions in higher codimensions are Λ n ξ n m ξ m a ξ a .
The two fully extended theories are still strongly equivalent and the map realizing the equivalence (namely the ones appearing in the diagram (40)) remain unchanged.
We have just to check that the actions are still preserved by the corresponding symplectomorphisms. In the bulk this has been proved in [CSS18, section 2.3]. The same argument can be adapted to higher codimension actions. The equations (ψ (k) ) * S (k) ΛBF * = S (k) ΛGR for k = 0, . . . 3 can also be verified by direct computation.