BV equivalence with boundary

An extension of the notion of classical equivalence of equivalence in the Batalin–Vilkovisky (BV) and Batalin–Fradkin–Vilkovisky (BFV) frameworks for local Lagrangian field theory on manifolds possibly with boundary is discussed. Equivalence is phrased in both a strict and a lax sense, distinguished by the compatibility between the BV data for a field theory and its boundary BFV data, necessary for quantisation. In this context, the first- and second-order formulations of nonabelian Yang–Mills and of classical mechanics on curved backgrounds, all of which admit a strict BV–BFV description, are shown to be pairwise equivalent as strict BV–BFV theories. This in particular implies that their BV complexes are quasi-isomorphic. Furthermore, Jacobi theory and one-dimensional gravity coupled with scalar matter are compared as classically equivalent reparametrisation-invariant versions of classical mechanics, but such that only the latter admits a strict BV–BFV formulation. They are shown to be equivalent as lax BV–BFV theories and to have isomorphic BV cohomologies. This shows that strict BV–BFV equivalence is a strictly finer notion of equivalence of theories.


Introduction
The notion of equivalence of field theories is one that can be found throughout physics. Such a concept is relevant and useful for various reasons. At a classical level, the equations of motions of a given theory might be easier to handle than other "equivalent" ones, even if they ultimately yield the same moduli space of solutions. Such reformulations often result in different and enlightening new interpretations of a given problem. Moreover, one theory might be better suited for quantisation than another, but the question of whether two classically equivalent theories result in the same quantum theory is in general still open. With this work, we attempt to take another step towards the answer.
The classical physical content of a given field theory is encoded in the set EL of solutions of the Euler-Lagrange equations. In the case where the theory in question also enjoys a local symmetry -encoded by a tangent distribution Dwe are interested in the moduli space of inequivalent solutions EL/D. Classical observables are then defined to be suitable functions on EL/D. Such a quotient is typically singular: defining a sensible space of functions over it becomes challenging and it is often more convenient to find a replacement; a problem best addressed within the Batalin-Vilkovisky (BV) formalism.
The BV formalism was first introduced in [BV83a; BV83b; BV84] as an extension of the BRST formalism [BRS76;Tyu08], named after Becchi, Rouet, Stora and Tyutin, used to quantise Lagrangian gauge theories in a way that preserves covariance. Around the same time, the Batalin-Fradkin-Vilkovisky (BFV) formalism was introduced, which deals with constrained Hamiltonian systems [BV77;BF83]. It was later noticed by various authors [McM84; Hen85; BM87; Dub87; FH90; Hen90a; McC94; Sta97; Sta98] that the aforementioned formalisms enjoy a rich cohomological structure. For example, a BV theory associates a chain complex to a spacetime manifold, the BV complex, which aims at a resolution of the desired space of functions over the quotient EL/D. In the case of the BFV formalism, one introduces the BFV complex [Sta97; Sch09a; Sch09b] as a resolution of the space of functions over the reduced phase space of a given constrained Hamiltonian system.
One can then address the question of equivalence of theories in the BV setting. Following the discussion above, a natural way of comparing two classical theories is through their BV cohomologies, also called classical observables, as done for example in [BBH95]. However, a BV theory comes equipped with several pieces of data other than the underlying dg-algebra structure (for example a symplectic structure and a Hamiltonian function) that one might want an equivalence relation to preserve. Finding the appropriate notion of "BV-equivalence" is thus a non-trivial open question. In [CSS18;CS22], a stronger notion of BV-equivalence is implemented, which requires all data to be preserved by a symplectomorphism. A nontrivial example of such equivalence is found between 3d gravity and (nondegenerate) BF theory. In [CCS21b] various alternative weaker notions of BV equivalence have been presented, which apply to higher dimensional formulations of General Relativity.
The BV and BFV approaches were linked by Cattaneo, Mnev and Reshetikhin in [CMR14], where the authors showed that a BV theory on the bulk induces a compatible BFV theory on the boundary, provided that some regularity conditions are met. The presence of a boundary will typically spoil the symmetry invariance of the BV data, encoded in the BV cohomology, but this failure will be controlled by the BFV data associated to the boundary. From this perspective, the regularity conditions can be seen as a compatibility condition between the BV complex on the bulk and the BFV complex on the boundary. Derived geometry [PTVV13] extends the above setting to algebraic geometry, even though currently only in the restricted setting of AKSZ theories. The induced boundary theory is in this case an example of derived intersection, [Cal15;CPTVV17]. As derived geometry mainly addresses classical problems, the nondegeneracy of the symplectic form is only required up to homotopy, which yields problems in the direction of quantisation.
On the other hand the BV-BFV approach [CMR14;CMR18] is especially successful since it allows for a quantisation procedure that is compatible with cutting and gluing. This has already been shown to work in various examples such as BF theory [CMR18; CMR20], split Chern-Simons theory [CMW17], 2D Yang-Mills theory [IM19] and AKSZ sigma models [CMW19].
This approach was first tested 1 on General Relativity in [Sch15]. For diffeomorphism invariant theories, the compatibility between bulk and boundary data becomes a non-trivial matter, and there are various cases where the regularity conditions necessary for the BV-BFV description fail to be met. Most notable are the examples of Palatini-Cartan gravity in (3+1) dimensions [CS19], Plebanski theory [Sch15], the Nambu-Goto string [MS22] and the Jacobi action for reparametrisation invariant classical mechanics [CS17]. On the other hand, the respectively classically equivalent Einstein-Hilbert formulation of gravity [CS16] in (3+1) dimensions and the Polyakov action [MS22] fulfill the BV-BFV axioms. The question of how one can go around these problems and construct a sensible BV-BFV theory for Palatini-Cartan gravity was addressed in [CCS21a ;CCS21b].
As not all field theories are suitable for a BV-BFV description, the lax approach to the BV-BFV formalism was proposed in [MSW20], which gathers the data prior to the step where the regularity conditions become relevant. This setting already allows us to construct the BV-BFV complex [MSW20], which is the adaptation of the BV complex to the case with boundary. Likewise, classical observables are contained in its cohomology. As such, the lax BV-BFV formalism offers a sensible way of comparing two field theories on manifolds with boundary, even if one does not have a strict BV-BFV theory.
In this paper, we provide an explicit method to lift classical equivalence to a (potential) BV equivalence, also in the presence of boundaries. This naturally introduces the notion of lax equivalence of BV theories on manifolds with boundary, which is in principle finer than BV equivalence. Our method is applied to the simple cases of classical mechanics on a curved background as well as to (nonabelian) Yang-Mills theory, where we explicitly show that the first-and secondorder formalisms are lax BV-BFV equivalent (and hence BV-quasi-isomorphic).
We then turn our attention to the main objective of this paper: the analysis of the classically-equivalent Jacobi theory and one-dimensional gravity coupled to matter (1D GR). These two models can be regarded as the one-dimensional counterparts of the Nambu-Goto and Polyakov string models respectively, and they both represent a reparametrisation-invariant version of classical mechanics. In [CS17] it was shown that while 1D GR satisfies the regularity conditions of the BV-BFV formalism, Jacobi theory produces a singular theory on the boundary, and a similar result was proven for their 2d string-theoretic analogues [MS22] which raises the question of the origin of this boundary discrepancy.
By comparing the BV and BV-BFV cohomologies of Jacobi theory and 1D GR, we find that, even though the two theories on manifolds with possibly non-empty boundary are lax equivalent, and hence their associated BV (and lax BV-BFV) complexes are quasi-isomorphic, the chain maps that connect the two theories do 1 Another approach to General Relativity by means of the BV formalism (without boundary) can be found in [Rej11;FR12b]. not preserve the regularity condition required by the strictification procedure (Theorem 3.5.7).
In other words, quasi-isomorphisms of lax BV-BFV complexes do not preserve strict BV-BFV theories, which then should be taken as a genuine subclass of BV theories: even in the best case scenario of two theories that are classically equivalent with quasi-isomorphic lax BV-BFV complexes, an obstruction to their strict BV-BFV compatibility distinguishes the two.
Indeed, consider two lax equivalent theories (Definition 2.6.3 -see e.g. the case described in Theorem 3.5.6) such that one of the two models fails to be compatible with the strict BV-BFV axioms (cf. Remark 2.5.6). In this case only one of the two admits a quantisation in the BV-BFV setting. Even if they both could ultimately admit a sensible quantisation, our result suggests anyway that they might have different quantisations in the presence of boundaries.
Another way of viewing our result is the following. Suppose we are given a lax BV-BFV theory that is not strict (Definition 2.5.5 and Remark 2.5.6). Can we find a quasi-isomorphic lax BV-BFV theory that is strictifiable? If so, we may think of the second theory as a good replacement for the first, suitable for quantisation with boundary.
We should stress that the enrichment of the BV complex by the de Rham complex of the source manifold (in the sense of local forms) has been the object of past research (see among all [BBH95]). The lax BV-BFV complex we consider coincides with their Batalin-Vilkovisky-de Rham complex; however, our notion of lax equivalence is different (Definition 2.6.3), as it requires the existence of chain maps that are quasi-inverse to one another and compatibile with the whole lax BV-BFV structure.
Crucially, our approach diverges from other investigations of local field theory that only look at pre-symplectic data. The strictification step is precisely the presymplectic reduction of such data, and where the obstruction lies. We are not aware of a viable quantisation procedure for pre-symplectic structures.
This paper is structured as follows: Section 2 is dedicated to a review of local Lagrangian field theory (Section 2.1), which is followed by the BV formalism (Section 2.2) and the BV-BFV and lax BV-BFV formalisms (Section 2.4). We will showcase several notions of equivalence in classical field theory, starting from Lagrangian field theory in Section 2.1, while the discussion of equivalence in the BV and lax BV-BFV cases can be found in Sections 2.3 and 2.6 respectively. Later in Section 3 we discuss our general procedure to prove lax equivalence between two theories (Section 3.1) and three examples of such equivalence, namely • first-and second-order formulations of classical mechanics on a curved background (Section 3.3); • first-and second-order formulations of (non-abelian) Yang-Mills theory (Section 3.4); • one dimensional gravity coupled to matter and Jacobi theory (Section 3.5).
Results and outlook: We present our notion of BV equivalence (Definition 2.3.1) for theories over closed manifolds and lax equivalence (Definition 2.6.3) in the case of manifolds with higher strata, and show that the latter implies the former for the respective bulk (codimension-0 stratum) BV theories (Theorem 2.6.9).
We then show lax equivalence for the aforementioned examples, in the sense that their lax BV-BFV data can be interchanged in a way that preserves their cohomological structure. In particular, we show that the respective BV-BFV complexes are quasi-isomorphic Most notably, this means that the boundary discrepancy present in the BV-BFV formulations of Jacobi theory and 1D GR found in [CS17] does not have a cohomological origin, and is rather to be interpreted as an obstruction in prequantisation.
We expect the procedure to be applicable to other relevant examples of BV-BFV obstructions such as the Nambu-Goto and Polyakov actions [MS22] and, for a more challenging one, Einstein-Hilbert and Palatini-Cartan gravity in (3+1) dimensions, whose extendibility as BV-BFV theories have been shown to differ in [CS16;CS19].
This obstruction, which bars certain theories from being quantisable in the BV formalism with boundary without additional requirements on the fields, suggests that, even assuming that some quantum theory exists for both models, they might differ. Alternatively, it might suggest that among various classically-and BVequivalent models, there is a preferred choice for models which are BV-BFV compatible. Either way, these results call for additional investigations in this direction.

Acknowledgements
We would like to thank G. Barnich, M. Grigoriev and M. Henneaux for instructive discussions on the topic of equivalence of field theories in the BV formalism, relevant to this work.

Field theories and equivalence
We start by presenting the field theoretical structures and objects used throughout this work, following [Del+99;And89]. Subsequently, we review the BV formalism for closed manifolds 2 [BV83a; BV83b; BV84] − see also [Hen90a; GPS95; Mne17] − and the BV-BFV formalism, its generalisation for manifolds with boundaries and corners [CMR14; CM20]. As some theories we consider are not compatible with the BV-BFV axioms, we revise the lax BV-BFV formalism [MSW20], which not only lets us study these cases, but presents a better stage for our discussions in the presence of boundaries and corners.
Moreover, this section is used to develop our notion of equivalence of field theories at every step of the way, first showcasing how we want to compare two classical field theories in Definition 2.1.6 and adapting these considerations to the BV and lax BV-BFV formalisms in Definitions 2.3.1 and 2.6.3 respectively.
2.1. Lagrangian field theories. Let M be a manifold of any dimension. In order to build a classical field theory on M we need a space of fields E, a local functional S called the action functional and local observables. In most cases, we can achieve such a construction by considering a (possibly graded) fibre bundle E → M over M and by defining the space of fields as its space of smooth sections E := Γ(M, E) with coordinates ϕ i . Local objects can then be regarded as a subcomplex of the de Rham bicomplex Ω •,• (E × M ), where "local" essentially means that these objects only depend on the first k derivatives of the fields ϕ i (or the k-th jet). Let us make this notion precise: Definition 2.1.1 ((Integrated) local forms [And89]). Let E → M be a (possibly graded) fibre bundle over M , E = Γ(M, E) its space of smooth sections, J k (E) the k-th jet bundle and {j k : E × M → J k (E)} the evaluation maps. We consider j ∞ as the inverse limit of these maps and construct the infinite jet bundle J ∞ (E) as the inverse limit of the sequence 2 For a discussion of the BV formalism in the setting of non-compact manifolds see [Rej11;FR12a]. For the extension of the BV-BFV framework to manifolds with asymptotic boundary see [RS21].
The bicomplex 3 of local forms on E × M is defined as where d H and d V are the horizontal and vertical differentials on the variational complex for J ∞ (E), respectively. Let α ∈ Ω •,• (J ∞ (E)). The differentials δ, d are defined through Elements of Ω 0,• loc (E × M ) will be called local functionals on E × M . Whenever the manifold M is compact, one can define the complex of integrated local k forms Ω • (E), as the image of M : Ω k,top loc (E × M ) → Ω k (E) with the (variational) differential 4 δ.
Remark 2.1.2 (On various notions of local forms). Notice that, in some fieldtheory literature (see e.g. [Del+99]), the term "local form" is often used to denote integrals over the manifold M of elements of Ω •,top loc (E × M ), which instead we call integrated local forms.
When M is not compact, integration comes with caveats. One can either consider compactly supported sections or adopt the point of view of [FR12a], where the Lagrangian density is tested against a compactly supported function. Alternatively, one can forgo integration and consider the following quotient In [And89,Page 21], the elements of Ω • loc (E) are called variational forms when endowed with the induced vertical differential 5 δ V .
Clearly, if M is closed, (Ω • (E), δ) is isomorphic (as a complex) to (Ω • loc (E), δ V ). Indeed, let f, g ∈ Ω •,top loc (E × M ) and define F := M f , G := M g, their respective integrals over M . Then F = G iff the difference f − g is d-exact where we used that M is closed in the last step. Hence, Ω • loc (E) can be taken as a replacement of integrated local forms in the noncompact case (assuming there still is no boundary).
If M has a non-empty boundary ∂M = ∅, these considerations no longer hold: boundary terms become relevant. If, on the local densities side we can work with Ω •,• loc (E × M ), we will see in Section 2.4 what the consequences of integrating over boundaries bring about in field theory.
In addition to the previous construction, we will extensively use the following type of vector field: It is possible to induce a differential coming from the variational bicomplex, by means of the interior Euler operator (see e.g. [And89]). We will not be concerned with the details of this construction.
We are now ready to define the notion of a classical field theory. We will assume for simplicity that M is compact, possibly with boundary: Definition 2.1.4. A classical field theory on M is a pair (E, S), consisting of a space of fields E = Γ(M, E) 6 and an action functional S ∈ Ω 0 (E).
Since S = L for some local form L, applying the variational differential on E to S is the same as applying δ, defined in Equation 1b, to L and integrating. This yields two terms: The term EL is an integrated local 1-form 7 on E, whose vanishing locus defines the Euler-Lagrange equations EL = 0. The space where these are satisfied is called the critical locus, the zero locus EL := Loc 0 (EL) ⊂ F , and its elements are called classical solutions. The term BT is a boundary term (i.e. an integral over ∂M , when not empty), which will be crucial for the construction of field theories on manifolds with boundary (cf. Section 2.5).
A further important aspect of field theories is the notion of (gauge) symmetries, which are transformations that leave the action functional S and the critical locus EL invariant. Infinitesimally, they can be described as follows: Definition 2.1.5. An infinitesimal local symmetry of a classical field theory (E, S) is given by a distribution D ⊆ T E, such that 8 Furthermore, we require D to be involutive on the critical locus Whenever local symmetries are present, the space of interest is not EL but rather the space of inequivalent configurations EL/D 9 , i.e. the space of orbits of D on the critical locus EL. Classical observables are then suitable functions over EL/D, whose space we denote by C ∞ (EL/D). Note that, as a quotient, EL/D is often singular and defining C ∞ (EL/D) is a non-trivial task. One way of handling this is to build a resolution of C ∞ (EL/D), by means of the Koszul-Tate-Chevalley-Eilenberg complex, also known as the BV complex (see Definition 2.2.4).
We are interested in analyzing to what extent two field theories are equivalent. Starting the discussion of equivalence in the setting of classical Lagrangian field theory, we consider the Definition 2.1.6. Let (E i , S i ), i ∈ {1, 2}, be two classical field theories with symmetry distributions D i . We say that (E i , S i ) are classically equivalent if 6 More generally, the space of fields is an affine space modeled on a space of sections; e.g., a space of connections. Even more generally, e.g., in the case of sigma models, one expands fields around a background field. It is the space of these perturbations that is a space of sections.
7 The integrand of EL is the pullback along j ∞ of a form of source type in the variational bicomplex, see [And89, Definition 3.5].
8 Notice that we want D to be a (generically proper) subspace of all vector fields that annihilate the action functional. We want it to be maximal, in the sense that all symmetries are considered except trivial ones, i.e. those that vanish on EL. As such it is not automatically a subalgebra. See [Hen90a, Section 1.3]. Observe that, although not necessary, one might want to restrict D to only (genuinely) local symmetries, meaning that we do not consider constant Lie group/Lie algebra actions at this stage. 9 By abuse of notation, we denote by D also the restriction of D to EL.
Remark 2.1.7. If two theories are classically equivalent, we have EL 1 /D 1 ≃ EL 2 /D 2 . If we have a model for the respective spaces of classical observables, they are isomorphic: This notion will be central in our discussion, and we will provide a refinement of it within the BV formalism, with and without boundary.
Remark 2.1.8. In certain cases we can find C 1 ⊂ E 1 , defined as the set of solutions of some of the equations of motion EL 1 = 0. Then, if we can find an isomorphism φ cl : C 1 → E 2 such that the theories are classically equivalent. This is a simple example of the situation in which two theories are classically equivalent because they differ only by auxiliary fields (see e.g. [BBH95]).

2.2.
Batalin-Vilkovisky formalism. The BV formalism is a cohomological approach to field theory, that allows one to characterise the space of inequivalent field configurations by means of the cohomology of an appropriate cochain complex. It turns out that it also provides a natural notion of equivalence of field theories, that also takes into account "observables" of the theory. In this setting, a classical field theory is described through the following data: • ω ∈ Ω 2 (F ) is an integrated, local, symplectic form on F of degree −1 (the BV form), • S ∈ Ω 0 (F ) is an integrated, local, functional on F of degree 0 (the BV action functional ), • Q ∈ X evo (F ) is a cohomological, evolutionary, vector field of degree 1, i.e.
[Q, Q] = 2Q 2 = 0, and [L Q , d] = 0, such that (2) The internal degree of F is called the ghost number and will be denoted by gh(·).
Remark 2.2.2. In principle, we only need to consider either S or Q, as they are related to one another through Equation (2), apart from the ambiguity of an additive constant in S. We will nonetheless regard them as separate data for later convenience, as we will see that introducing a boundary spoils Equation (2).
As Q is cohomological, its Lie derivative L Q is a differential on Ω • loc (F ), since gh(L Q ) = 1 and 2L 2 Q = [L Q , L Q ] = L [Q,Q] = 0. In this context, L Q -cocycles are interpreted as (gauge-)invariant local forms.
Remark 2.2.3. It is easy to gather that both ω and S are L Q -cocycles by applying δ and ι Q to Equation (2) respectively. We have 10 For simplicity, in this note we assume that the Grassmann parity of a variable is equal to the parity of its Z-degree. This is okay as long as we only consider theories without fermionic physical fields.
where (·, ·) is the Poisson bracket induced by ω. Equation (3b) is known as the Classical Master Equation [BV84;Sch93], and encodes the property that S is gauge invariant. In particular, Equations (3) mean that we have the freedom to perform the transformations ω → ω + L Q (. . . ) and S → S + L Q (. . . ), as long they preserve Equation (2).
Definition 2.2.4. We define the BV complex of a given BV theory F as the space of integrated local forms on F endowed with the differential L Q where the grading on BV • is given by the ghost number. Its cohomology will be denoted by H • (BV • ) and called the BV cohomology.
While the BV complex BV • consists of inhomogeneous local forms (inhomogeneous also in ghost number), its 0-form part 11 BV • 0 ⊂ BV • is of interest as it is a resolution of D-invariant functionals on EL or, when the quotient is nonsingular, functionals on EL/D in the sense that the BV cohomology is given by 12 [Hen90a; . Example 2.2.5 (Lie algebra case [BV84], see also [Hen90a;Mne17]). In this paper we will only consider examples which enjoy symmetries that come from a Liealgebra action. Let (E, S) be a classical field theory over a closed manifold M with a symmetry on E given by the action of a Lie algebra (g, [·, ·]). We can build a BV theory as follows: choose the space of fields to be with local coordinates Φ i = (ϕ j , ξ a ) on the base E × Ω 0 (M, g)[1] and Φ † i = (ϕ † j , ξ † a ) on the fibers. Usually one calls ϕ j the fields, ξ a the ghosts 13 and Φ † i the antifields. Note that the ghost numbers are related by gh(Φ i ) + gh(Φ † i ) = −1 due to the -1 shift on the fibers. We take the BV form to be the canonical symplectic form on F where ·, · is a bilinear map with values in Ω •,top loc (M ). In the case of a Lie algebra action, the cohomological vector field Q decomposes into the Chevalley-Eilenberg differential γ and the Koszul-Tate differential δ KT The action of γ is defined on the fields and ghosts as where v i a are the fundamental vector fields of g on F . In turn, δ KT acts as 11 In the literature the terminology "BV complex" is used to denote BV • 0 . We use the same name for BV • as it is the natural extension in the present setting.
12 Counterexamples to this scenario have been observed [Get16;Get17]. In local field theory, the request that the BV complex be a proper resolution of the moduli space of the theory is generally too strong. Hence, we do not insist on the vanishing of negative cohomology. 13 In the case of Yang-Mills theory, the ghost field will be denoted as c.
The BV action functional can then be constructed as an extension of the classical action functional and Q(·) = (S, ·) can be used to compute the full form of QΦ † i . The data (F , ω, S, Q) form a BV theory.
2.3. Equivalence in the BV setting. We now have all the necessary tools to develop a notion of equivalence in the BV formalism. We are interested in comparing the BV data and cohomology H • (BV • i ) of two BV theories F i , i ∈ {1, 2}. We recall that a quasi-isomorphism is a chain map between chain complexes which induces an isomorphism in cohomology. In this spirit we define: Definition 2.3.1. Two BV theories F 1 and F 2 are BV-equivalent if there is a (degree-preserving) map φ : F 2 → F 1 that induces a quasi-isomorphism φ * : BV • 1 → BV • 2 of BV complexes, such that φ * preserves the cohomological classes of the BV form and BV action functional as A BV equivalence is called strong iff φ is a symplectomorphism that preserves the BV action functionals.
Remark 2.3.2. If F 1 , F 2 are BV-equivalent, we can find a morphism ψ : F 1 → F 2 , such that its pullback map ψ * is the quasi-inverse of φ * . In particular, the composition maps . This is equivalent to the existence of two maps h χ : Furthermore, note that applying ψ * to Equation (5) yields Let us now explore some direct implications of Definition 2.3.1, in particular that the transformation of ω i and S i are not independent: Proposition 2.3.3. Rewrite Equations (5) and (6) as Moreover, Equation (7) is satisfied if Proof. Applying φ * to ι Q1 ω 1 = δS 1 yields and analogously L Q1 (ι Q1 ρ 1 + δσ 1 ) = 0.
Remark 2.3.5. In the literature there exists another notion of equivalence of BV theories, based on what is usually called elimination of (generalised) auxiliary fields or reduction of contractible pairs (see e.g. [Hen90b; BBH95] and [BG11] for a review). When two theories differ by auxiliary fields content, they have the same BV cohomology. In Section 3.2 we show how the presence of auxiliary fields leads to the process of elimination of cohomologically contractible pairs by explicitly constructing chain maps that are quasi inverse to one another and homotopic to the identity. Hence, theories that differ by auxiliary fields/contractible pairs are BV equivalent in the sense of Definition 2.3.1.

2.4.
Field theories on manifolds with higher strata. The BV formalism can be extended to the case where the underlying manifold M has a non-empty boundary ∂M = ∅, as presented in [CMR14]. This construction relies on the BFV formalism introduced in [BF83], see also [Sta97;Sch09a;Sch09b].
[Q ∂ , Q ∂ ] = 0, and [L Q ∂ , d] = 0 such that Q ∂ is the Hamiltonian vector field of S ∂ We call ω ∂ , S ∂ the boundary form and boundary action functional respectively.
Definition 2.4.2. A BV-BFV theory over a manifold M with boundary ∂M is given by the data where (F ∂ , ω ∂ , S ∂ , Q ∂ ) is an exact BFV theory over Σ = ∂M and π : F → F ∂ is a surjective submersion such that and Q • π * = π * • Q ∂ .
Note that the failure of the structural BV forms to be L Q -cocycles is controlled by (boundary) BFV forms. In particular, Equation (10b) means that S fails to be gauge invariant, and the right hand side can be related to Noether's generalised charges [RS21]. Furthermore, the CME no longer holds. Instead we have the modified Classical Master Equation [CMR14] 1 2 ι Q ι Q ω = π * S ∂ .
2.5. Inducing boundary BFV from bulk BV data. It is important to emphasize how one can try to construct a boundary theory F ∂ from a BV theory F, since there might be obstructions. The problem we want to address is that of inducing an exact BFV theory on the boundary ∂M , starting from the BV data assigned to the bulk manifold M . Defineα as:α By restricting the fields of F (and their normal jets) to the boundary ∂M we can define the space of pre-boundary fieldsF ∂ and endow it with a pre-boundary 2-form ω = δα. Usuallyω turns out to be degenerate. In order to define a symplectic space of boundary fields, one then has to perform symplectic reduction, see, e.g., [Sil08]. Let kerω = {X ∈ X(F ∂ ) |ι Xω = 0 } and set Since we are taking a quotient, nothing guarantees that F ∂ is smooth, but we want to assume that this is the case. However, a necessary condition for smoothness is that kerω has locally constant dimension, i.e. it is a subbundle of TF ∂ . As we will see, this condition is not always satisfied, namely that there is a unique symplectic form ω ∂ such that π * ω ∂ =ω, and a unique cohomological vector field Q ∂ such that Q • π * = π * • Q ∂ . We assume (although this may not be true in general) that there is a 1-form α ∂ such that π * α ∂ =α. Note that, in this case, α ∂ is unique and ω ∂ = δα ∂ . See [CM20] for details. However, for F ∂ smooth, we have the surjective submersion π : F → F ∂ . Consider now Definition 2.5.1. The graded Euler vector field E ∈ X evo (F ) is defined as the degree 0 vector field which acts on local forms of homogeneous ghost number as The cohomological vector field Q ∂ is actually Hamiltonian and the corresponding boundary action functional can be computed as [Roy07] The data F ∂ = (F ∂ , ω ∂ , S ∂ , Q ∂ ) define an exact BFV manifold over the boundary ∂M . For completeness, we also define the pre-boundary action functionalŠ := π * S ∂ . Pulling back Equation (13) via π * yieldš Note that by taking Equations (11) and (14), we ensure that the data (α,Š) can always be defined, even if the quotient in Equation (12) does not yield a smooth structure.
Remark 2.5.2. The procedure we just presented can be repeated in case that the manifold M not only has a boundary but also corners (higher strata), as presented in [CMR14;MSW20]. If this is possible up to codimension n, then we call the theory a n-extended (exact) BV-BFV theory.
Remark 2.5.3. The quantization program introduced in [CMR18] relies on the BV-BFV structure of a given classical theory. As such, even if two theories are classically equivalent, only one might turn out to have a BV-BFV structure and so be suitable for quantization, as we now explore in the example of the Jacobi theory and 1D GR.
Remark 2.4.3 and Section 2.5 discuss two potential roadblocks for our construction of equivalence in the presence of boundaries and corners (and more generally codimension-k strata). First, to extend the notion of equivalence discussed in Section 2.3 to the case with higher strata, we wish to capture the possibility of local forms being L Q -cocycles up to boundary terms, as is the case with ω and S. This is the problem of descent, where we enrich the differential L Q by the de Rham differential on M . The second big problem one encounters is that not all BV theories satisfy the regularity requirement necessary to induce compatible BV-BFV data. In order to describe such theories as well we will relax our definitions.
In order to do this, we turn to a "lax" version of the BV-BFV formalism [MSW20]. We will work with local forms on F × M with inhomogeneous form degree on M , namely κ • ∈ Ω p,• loc (F × M ), and use the codimension to enumerate them, as it makes the notation less cumbersome and more intuitive, i.e. κ k denotes the (top − k)-form part of Remark 2.5.4. What we call "lax" BV-BFV formalism is a rewriting of known approaches to local field theory in the BV/BRST formalism such as [BBH95; Bra97; BBH00; Gri11; Sha16]. We use the term "lax" to contrast it with the "strict" version given by the BV-BFV formalism proper.
This should be compared to the standard BV-BFV formalism (extended to codimension k [CMR14]), which instead looks at Ω • loc (F (k) ) with F (k) an appropriate space of codimension-k fields. In other words, we describe the BV-BFV picture presented above in terms of densities instead of integrals (cf. Remark 2.1.2), and forfeiting the symplectic structure at codimension k. This setting allows us to phrase equivalence with higher strata in a cohomological way, and it collects all the relevant data before performing the quotient in Equation (12), thus temporarily avoiding potential complications. 14 The definitions that we work with rely on the lax degree 15 #(·), which describes the interplay between the co-form degree on M and the ghost number. Let fd M (·) denote the form degree on M . The lax degree is defined as the difference of the ghost number gh(·) and the co-form degree cfd M (·) : In particular, if an inhomoegeneous local form has vanishing lax degree, then the codimension of its homogeneous components corresponds to their respective ghost number. Most notably, this will be the case for the Lagrangian density. We will 14 A similar idea is contained in the work of Brandt, Barnich and Henneaux [BBH95], but without the structural BV-BFV equations. 15 In [MSW20] the authors denote the lax degree by total degree. use the total degree for computations, which for elements in Ω •,• loc (E × M ) is given by | · | = gh(·) + fd M (·) + fd F (·), where fd F (·) is the form degree on F .
Remark 2.5.6 (Strictification of lax data). Let M • be the interior (bulk) of M .
(1) If M = M • is a closed manifold, then we can assign a BV theory F to M • from a lax BV-BFV theory F lax by choosing 16 and restricting Q to F . (2) Similarly, if M is a compact manifold with boundary, the pre-BFV data on ∂M may be induced by settinǧ and restricting Q toF ∂ . When pre-symplectic reduction w.r.t.ω = δα = M ∂ ̟ 1 is possible [CMR14], we can define the space of boundary fields F ∂ :=F ∂ /ker(ω ♯ ). Together with the bulk data presented above, this produces a BV-BFV theory.
The procedure is analogous for higher codimensions: if M (k) denotes the kthcodimension stratum, we can induce a Hamiltonian dg manifold of fields in codimension k by performing pre-symplectic reduction of Notice that pre-symplectic reduction might fail to be smooth, resulting in an obstruction to strictification. When there are no obstructions to the pre-symplectic reduction, this procedure yields an n-extended BV-BFV theory (cf. Remark 2.5.2), and we have an n-strictification of a lax BV-BFV theory. (For more details we refer to [MSW20].) It is crucial to observe that this step can fail [CS19; CS17; MS22]. Unlike the latter, a lax BV-BFV theory does not require working with symplectic structures at higher codimensions ≥ 1. This means that lax data allow us to extract some information about the higher codimension behaviour of the field theory but, as we will see, the fact that a theory is strictifiable at a given codimension yields a refinement of the notion of BV equivalence. 16 We denote by F lax | M • (resp. F lax | M ∂ ) the restriction of fields to the interior (resp the boundary stratum, where we also restrict normal jets) of M , seen as section of a fibre bundle (resp. the tangent bundle to the induced bundle).
Remark 2.5.7. At codimension ≥ 1, it is sufficient to know θ • in order to compute L • . Applying ι E to Equation (15a) yields We can then compute L k at codimension k ≥ 1 by using gh(L k ) = cfd M (L k ) = k: Lemma 2.5.8 ([CMR14; MSW20]). The following equations hold for a lax BV-BFV theory: Remark 2.5.9. Equations (17) are the density versions of Equations (10). Comparing the two versions, we see that boundary terms are now encoded as d-exact terms, instead of objects in the image of π * . Note that ̟ • is a cocycle of the differential (L Q − d) and that L • is so whenever In the lax BV-BFV formalism, the relevant differential will no longer be L Q , as we want to take the boundary configurations into account. Instead, we want to consider a cochain complex of local forms on F lax × M with differential L Q − d, which describes the interplay between gauge invariance and boundary terms: Definition 2.5.10 ([BBH95; MSW20]). The BV-BFV complex of a lax BV-BFV theory F lax is defined as the space of inhomogeneous local forms on F lax × M endowed with the differential (L Q − d) where the grading of (BV-BFV) • is given by the lax degree. We will denote its cohomology by H • ((BV-BFV) • ) and call it the BV-BFV cohomology.
Remark 2.5.11. The cocycle conditions for an inhomogenoeous local form Such equations are of interest since their solutions produce classical observables, i.e. local functionals (i.e. p = 0) which belong to H 0 (BV • ). Let γ k denote a (dim M − k)-dimensional closed submanifold of M . We can then construct a classical observable by integrating O k over γ k since As such, comparing the BV-BFV cohomologies of two lax theories offers a natural way of comparing their spaces of classical observables.
2.6. Equivalence in the lax setting. Before adapting our notion of equivalence to the case when a boundary and corners are present, let us define f -transformations, which encode the facts that eventually (i) we are interested in the 2-forms ̟ • (and not in their potentials θ • ) and (ii) Lagrangian densities will be integrated (so total derivatives become irrelevant). The two issues are actually related.
Remark 2.6.2. Note that an f -transformation preserves Equations (15) since where we used [δ, d] = 0. Hence, we will also allow this kind of freedom in our definition of equivalence.
In the following, we will denote the vertical differentials on F lax i by δ and the horizontal (de Rham) differentials on M i by d.
Definition 2.6.3 (Lax equivalence). We say that two lax theories F lax 1 and F lax 2 are lax equivalent if there are two morphisms of graded manifolds φ : between the BV-BFV complexes, such that φ * and ψ * are quasi-inverse to each other and transform (θ Remark 2.6.4. Similarly to the bulk case, in order to show that the composition maps , are the identity when restricted to the respective cohomologies, one needs to find two maps h χ : 17 We restrict ourselves to the gh(φ) = gh(ψ) = 0 case as this will be the relevant one in our examples.
(1) As gh(φ) = 0, φ * commutes with the Euler vector fields L Ei : . By counting degrees we see that both sides have ghost number k at codimension k, therefore for k ≥ 1 one can use this equation where we used ι Q2 ̟ • 2 = δL • 2 +dθ • 2 , δ 2 = 0 and the fact that f -transformations preserve Equation (15a). Note that this condition holds automatically for condimention higher than zero due to ζ k 2 = ι Q2 β k 2 . To see what Equation (19) implies at codimention zero, first note that The computations are analogous for i = 1.
Remark 2.6.6. The previous lemma means that there is a redundancy in our definition of lax equivalence, as the transformation of L • i at codimension ≥ 1 can be determined through the transformation of θ • i . In particular, when computing explicit examples one only needs to check if we have the right transformation for θ Remark 2.6.7. Our definition of lax equivalence directly implies that the local i ensures that the second condition from Proposition 2.6.5 is satisfied.
The proof is analogous to the proof of Proposition 2.3.4.
. We need to check if: (1) φ * , ψ * are chain maps w.r.t. the BV complexes, (2) the cohomological classes of ω i and S i are mapped into one another, To prove these, we simply need to integrate the various conditions over the bulk M . For the chain map condition we have We can compute the transformations of ω 1 and S 1 in a similar way In the same manner implying that χ * , and analogously λ * , are also homotopic to the identity in BV • i and as such the identity when restricted to H • (BV • i ), meaning that the BV complexes are quasi-isomorphic.

Examples
This section is dedicated to the explicit computation of lax BV-BFV equivalence in three different examples. We start by presenting the general strategy in Section 3.1. We then look at the examples of classical mechanics on a curved background and (non-abelian) Yang-Mills theory in Sections 3.3 and 3.4 respectively. Subsequently we turn our attention to the classically-equivalent Jacobi theory and one-dimensional gravity coupled to matter (1D GR) in Section 3.5, and show that they are lax BV-BFV equivalent, despite their different boundary behaviours w.r.t. the BV-BFV procedure. Furthermore, we show that the chain maps used to prove lax BV-BFV equivalence spoil the compatibility with the regularity condition for the BV-BFV procedure in the case of 1D GR.
3.1. Strategy. We shortly demonstrate our strategy to show explicitly that two theories F lax i are lax BV-BFV equivalent. In practice, we need two maps φ * , ψ * between the BV-BFV complexes BV-BFV (1) are chain maps, are quasi-inverse to one another. In order to check the first property we note that, as pullback maps, φ * and ψ * automatically commute with the de Rham differentials δ and d. Thus it suffices to show that φ * , ψ * are chain maps w.
, as this together with the fact that they commute with δ then implies that they are chain maps w.r.t. (L Qi − d) on BV-BFV • , resulting in the following Lemma: Showing the second property is a matter of computation. For the third property, we shortly present our strategy to show that χ * = ψ * • φ * is the identity in cohomology. The same procedure can then be applied to We start by constructing a homotopy between χ * and id 1 , by finding an evolutionary vector field R 1 ∈ X evo (F 1 ) with gh(R 1 ) = #(R 1 ) = −1 and defining a one-parameter family of morphisms of the form , such that χ * s=0 = id 1 and lim s→∞ χ * s = χ * . Note that choosing s = − ln τ gives the usual definition of homotopy with τ ∈ [0, 1], i.e. a continuous map F (τ ) := χ * (− ln τ ) satisfying F (0) = χ * and F (1) = id 1 . We will nonetheless work with the parameter s to keep the calculations cleaner, changing when necessary. Furthermore, we can simplify the term in the exponent by setting which results in Lemma 3.1.2. The Lie derivative L D1 commutes with the differential (L Q1 − d).
Proof. Since R 1 and Q 1 are evolutionary vector fields, we directly have [L D1 , d] = 0. To see that L D1 commutes with L Q1 we compute where we have used the graded Jacobi identity and [Q 1 , Q 1 ] = 0. Thus proving the statement.
We can then determine the map h χ by rewriting the RHS of Equation (20) as resulting in the following Lemma: satisfies Equation (20).
If h χ converges on BV-BFV • 1 , then χ * will be the identity in the BV-BFV cohomology H • (BV-BFV • 1 ) as desired. For this last step, the next Lemma will be useful: , such that we integrate over a compact interval instead of over R ≥0 . Performing this transformation results in Assuming that we can show that h χ κ converges on BV-BFV • 1 . Let first κ = f be a local functional.
where we used that χ * s = e sLD 1 = e − ln(τ )LD 1 is a morphism in the last equality. The integral over the first integrand is finite by assumption and the second integrand where J is a multiindex raging over the fields and their jets, and ν ∈ Ω • (M ) is a form on M . We then have The terms in the brackets {·} are just the integrands of h χ f and h χ ϕ J , which converge. Since the other terms e − ln(τ )LD 1 ϕ J = e sLD 1 ϕ J , e − ln(τ )LD 1 f = e sLD 1 f are well-defined ∀τ ∈ [0, 1] and we are integrating over a compact interval, the integral converges and h χ is well-defined on BV-BFV • 1 . 3.2. Contractible pairs. The simplest example we can discuss is when the cohomologically trivial fields are nicely decoupled from the rest. This follows the procedure of [Hen90b], which we explicitly embed in our framework by constructing suitable chain maps to fit Definition 2.3.1 (see also [BBH95] and [BGST05; BG11; Gri11]). Namely, we have an action of the form where ( , ) is some constant nondegenerate bilinear form on the space V of the v fields and S 2 is a solution of the master equation (w.r.t. the fieldsã,ã † ). We want to compare this theory with the one defined by S 2 [a, a † ], where we remove the tilde for clarity of the notation. The cohomological vector field Q 1 of S 1 acts on the fieldsã,ã † as Q 1 . In addition we have Lemma 3.2.1. The composition map λ * = φ * • ψ * is the identity, while the composition map χ * = ψ * • φ * acts as and is homotopic to the identity.
Proof. One can directly check that λ * is the identity. In order to show that χ * is the identity in cohomology we define a family of maps χ * s = e sLD 1 , where D 1 = [Q 1 , R 1 ], and show lim s→∞ χ * s = χ * . We choose R 1 to act as We can then compute and similarly χ * ∞ v † = 0. On the other hand, D 1 a = D 1 a † = 0 =⇒ e sLD 1 a = a, e sLD 1 a † = a † , ∀s, so that lim s→∞ χ * s ≡ χ * .
Furthermore the map h χ converges on all the fields, as h χ a = h χ a † = h χ v † trivially and A direct consequence of this Lemma, together with the facts that φ * S 1 = S 2 and φ * ω 1 = ω 2 for the canonical BV forms, is Theorem 3.2.2. The BV theories defined by S 1 and S 2 are BV equivalent.
3.2.1. More general contractible pairs. As in [Hen90b] we may consider a situation where S 1 depends on fieldsã, w,ã † ,w † , where (w, w † ) are not a contractible pair on the nose but satisfy the condition that has a unique solution w = w(ã,ã † ). We can then get closer to the previously discussed case by defining 18 We have indeed that Q 1 v † = v and Q 1 v = 0. Moreover, the above condition implies that the change of variables (w, w † ) → (v, v † ) is invertible (near w † = 0, or everywhere if w † is odd) and that the submanifold defined by the constraints v = 0 and v † = 0 is symplectic.
The above strategy then works, in the absence of boundary, with some modifications. Namely, the fact that now v and v † are not Darboux coordinates requires modifying the map of Equations (21). In turn, the transformation R 1 will get a nontrivial action on the fields (a, a † ).
All the examples we discuss below belong to this class of contractible pairs. As we will see, the modifications required in the case of classical mechanics and Yang-Mills theory are minimal, whereas those required in the case of 1D parametrisation invariant theories are more consistent-due to the fact that pairing of the v fields will depend on the a fields. In addition, in all the examples we show how to extend this construction to lax theories in order to encompass the presence of boundaries.
3.3. Classical mechanics on a curved background. We start by discussing the example of classical mechanics on a curved background as a warm-up exercise. We take the source manifold to be a time interval I = [a, b] ⊂ R and some smooth Riemannian manifold (M, g) as target. We will denote time derivatives with a dot and use tildes to distinguish fields between the different formulations of the theory.
We can formulate the theory by considering a "matter" fieldq ∈ F 2 := C ∞ (I, M ), and the metric tensor on the target will depend on the mapq. We introduce the shorthand notationg :=g(q). The classical action functional is given by This is usually called the second-order formulation.
On the other hand, we can phrase the theory in its first-order formulation, by considering again a map q : I → M , together with an "auxiliary" field 19 18 We may also think of this construction as the semiclassical approximation of a BV pushforward [CMR18, Section 2.2.2] that gets rid of the (w, w † ) variables. Indeed, the above condition may be read as the statement that setting w † to zero is a good gauge fixing. 19 Obviously the pair (q, p) is a map from I to T * M .
p ∈ C ∞ (I, q * T * M ) and the classical action functional where h := g −1 denotes the inverse of the target metric. For ease of notation, we will introduce the musical isomorphisms We recall the rather obvious and well-known Proof. Following Definition 2.1.6, we start by solving the EL equation of the firstorder theory corresponding to the auxiliary field p, we have which results in Let C 1 be the set of such solutions and define the map φ cl : hence the two formulations are classical equivalent.
Both of these theories can be extended to the lax BV-BFV formalism. Note that, as there are no gauge symmetries in these models, there is no need to introduce ghost fields. We start with the second-order formulation: , which are given by θ • 2 = q † , δq dt + g ♭ (q), δq , and the cohomological vector field Q 2 ∈ X evo (F lax 2CM ) where 20 ∂g := δg δq , defines a lax BV-BFV theory.
Proof. Again we need to check Equations (15) at codimension 0 and 1. Explicitly we have As in the second-order theory, the Lagrangian only has a top-form component. The only non-trivial equation is ι Q ̟ 0 We now present the main theorem of this section, together with an outline of its proof. 21 The computational details and the various lemmata are presented afterwards.
Theorem 3.3.4. The lax BV-BFV theories F lax 1CM and F lax 1CM of the first-order and second-order formulations of classical mechanics with a background metric are lax BV-BFV equivalent.
Proof. We need to check all the conditions from Definition 2.6.3. The existence of two maps φ, ψ with the desired properties is presented in Lemmata 3.3.5 and 3.3.6 respectively, where we also show that the pullback maps φ * , ψ * are chain maps w.r.t. the BV-BFV complexes BV-BFV • i , and that they map (θ • i , L • i ) in the desired way.
Furthermore, we need to show that the respective BV-BFV complexes are quasiisormophic. The composition map λ * = φ * • ψ * is shown to be the identity in Lemma 3.3.7. In Lemma 3.3.8, we prove that the composition map χ * = ψ * • φ * , is homotopic to the identity by following the strategy presented in Section 3.1.
In Lemma 3.3.9 we demonstrate that χ * is the identity in cohomology by showing that the map Lemma 3.1.3) converges, therefore proving that the two lax BV-BFV theories in question have isomorphic BV-BFV cohomologies and thus that they are lax BV-BFV equivalent.
Let us now look at the computations in detail. We start with the chain maps: Its pullback map φ * is a chain map w.r.t. (L Qi − d) and maps the lax BV-BFV data of the first-order theory as For a similar conclusion to the one presented here, see [Gri11, Section 3.2]. Our method allows to additionally construct explicit chain maps that implement the quasi-isomorphism.
Together with Proposition 3.1.1, this shows that φ * is a chain map w.r.t. (L Qi − d).
In turn, φ * acts on θ Lemma 3.3.6. Let ψ : F lax 1CM → F lax 2CM be the map defined through Its pullback map ψ * is a chain map w.r.t. (L Qi − d) and maps the lax BV-BFV data of the second-order theory as Proof. The only non-trivial calculation that is needed to check if ψ * is a chain map w.
and is homotopic to the identity.
We see that for all k ≥ 0, so that All in all, the homotopy χ * s takes the form χ * s q = q, χ * s p = e −s p − (e −s − 1)g ♭ (q), χ * s p † = e −s p † , and clearly satisfies lim s→∞ χ * s = χ * .
Lemma 3.3.9. The map χ * is the identity in cohomology.
Proof. We have to check if the map h χ converges on F lax 1CM , namely As {q, q † , p † } ∈ ker R 1 we have For h χ p † we compute thus, by Proposition 3.1.4, h χ converges on BV-BFV • 1 and χ * is the identity in cohomology.

Yang-Mills theory.
We now look at the example of (non-abelian) Yang-Mills theory. Let (M, g) be a d-dimensional (pseudo-)Riemanian manifold and G a connected Lie group with Lie algebra (g, [·, ·]), endowed with an ad-invariant inner product, which for ease of notation will be denoted by means of an invariant trace operation 22 Tr[·]. As we consider two formulations of Yang-Mills theory, we will use tildes to distinguish the fields between the two. We point out that an alternative proof of the equivalence of first-and second-order formulations of Yang-Mills theory has been given in [RZ18] using homotopy transfer of A ∞ -structures. We will give here an argument that is different on the surface, but which is compatible to their results. However, we stress that our analysis also includes a comparison of the boundary data of the first-and second-order formulations.
We can phrase the theory by considering connection 1-formsÃ ∈ Ω 1 (M, g), with curvatureFÃ, and the classical action functional This is often known as the second-order formulation.
Alternatively, one can phrase the theory in its first-order formulation, by considering an additional "auxiliary" field B ∈ Ω d−2 (M, g) and the classical action functional where ε s = ±1 denotes the signature of g.
Proposition 3.4.1. The first-and second-order formulation of Yang-Mills theory are classically equivalent.
Proof. Solving the EL equations of the first-order theory w.r.t. the auxiliary field B gives Let C 1 be the set of solutions of F A = ε s ⋆ B, or equivalently where we used ⋆ 2 = ε s (−1) k(d−k) when acting on k-forms, and let φ cl : C 1 → F 2 be the map defined through φ * clÃ = A. Then showing that the two theories are classically equivalent.
Both first-and second-order formulations of Yang-Mills theory can be extended to lax BV-BFV theories as follows. As the symmetries of the theory are given by a Lie algebra g, we can follow the construction of Example 2.2.5. 22 For a better nonperturbative behavior one usually requires G to be compact, in which case one uses the Killing form as the invariant inner product. This is the motivation for using the trace notation.
In the first-order formulation we have: and the cohomological vector field Q 1 ∈ X evo (F lax 1Y M ) defines a lax BV-BFV theory.
We now present the main theorem of this section, together with an outline of the proof. The computational details and the various required Lemmata are presented in the Appendix A. Proof. We need to check all the conditions from Definition 2.6.3. The existence of two maps φ, ψ with the desired properties is presented in Lemmata A.1.1 and A.1.2 respectively, where we also show that the pullback maps φ * , ψ * are chain maps w.r.t. the BV-BFV complexes BV-BFV • i , and that they map (θ • i , L • i ) in the desired way. Specifically, φ : and maps the lax BV-BFV data of the first-order theory as and maps the lax BV-BFV data of the second-order theory as in accordance with our notion of lax BV-BFV equivalence. Note that f 1 1 = f 2 1 = 0. Furthermore, we need to show that the respective BV-BFV complexes are quasiisormophic. The composition map λ * = φ * • ψ * is shown to be the identity in Lemma A.1.3, which follows directly from φ * B † = 0. In Lemma A.1.4, we prove that the composition map χ * = ψ * • φ * , which has the explicit form is homotopic to the identity by constructing the morphism χ * s = e sLD 1 with D 1 = [R 1 , Q 1 ], where R 1 is chosen to act as The homotopy is explicitly given by and fulfils lim s→∞ χ * s = χ * . In Lemma A.1.5 we demonstrate that χ * is the identity in cohomology by showing that the map therefore proving that the two lax BV-BFV theories in question have isomorphic BV-BFV cohomologies and thus that they are lax BV-BFV equivalent.
3.5. 1D reparametrisation invariant theories. In this section we compare two one-dimensional reparamentrisation invariant theories, namely Jacobi theory, which one can think of as classical mechanics at constant energies, and one-dimensional gravity coupled to matter (1D GR). For an in-depth discussion of these theories we refer to [CS17]. We recall that the motivation to investigate the equivalence of these two theories is that, even though they are classically equivalent, 1D GR is compatible with the BV-BFV procedure while the Jacobi theory is not and yields a singular boundary structure. Firstly, this raises the question whether this boundary discrepancy is reflected at a cohomological level. Secondly, this discrepancy in the boundary behaviour is also present in the classically equivalent Einstein-Hilbert gravity and Palatini-Cartan gravity in (3+1) dimensions, where the latter is incompatible with the BV-BFV procedure. Our hope is that the comparison and analysis of these toy models might shed light in the question of equivalence of the (3+1) dimensional theories. We take the base manifold to be a closed interval on the real line M = I = [a, b] ⊂ R with coordinate t for both theories, which should be interpreted as a finite time interval.
In the case of Jacobi theory, we consider a matter fieldq ∈ Γ(R n ×I) = C ∞ (I, R n ) with mass m. The kinetic energy is taken to be T (q) = m 2 q 2 where · is the Euclidean norm on R n andq = ∂ tq is the time derivative ofq. Let E denote a parameter and V (q) a potential term. We do not assume E = T (q) + V (q). The Jacobi action functional takes the form To see that S J is parameterisation invariant, note that writing ds 2 = 2m(E − V ) dq 2 lets us interpret the Jacobi action functional as the length of a path in the target space R n with metric ds 2 . As such the symmetry group of Jacobi theory is the diffeomorphism group of the interval Diff(I), i.e. the reparameterisations of I. The critical locus of S J is then given by the geodesics of the metric ds 2 , which are the trajectories of classical mechanics with an arbitrary parameterisation [CS17]. Imposing E = T (q) + V (q) allows us to recover the standard parameterisation. We set V (q) = 0 for the rest of the discussion. The EL equations can be shown to have the form which are singular forq = 0. As such, the space of fields for Jacobi theory is not C ∞ (I, R n ) but rather We can then interpret Jacobi theory as classical mechanics at constant energies where the solutions do not have turning points, i.e. points in which the first derivative vanishes.
In the case of 1D GR [CS17] we also consider a metric field g ∈ Γ(S 2 + T * I) as a non-vanishing section of the bundle of symmetric non-degenerate rank-(0, 2) tensors over I. For simplicity, we write g = g dt 2 and work with the component g ∈ C ∞ (I, R >0 ). The space of fields is given by The conditionq = 0 in F J is strictly speaking not necessary in the 1D GR case, but we are ultimately interested in comparing 1D GR with the Jacobi theory and therefore impose it for consistency. In this picture, we can interpret 1D GR as an extension of Jacobi theory. We consider the action functional Note that the Ricci tensor vanishes in 1D and hence the Einstein-Hilbert term is absent. The first term in S GR is simply the matter Lagrangian for vanishing potential in the presence of a metric field and the second is a cosmological term. As such we interpret the parameter E as a cosmological constant. Since we are integrating over the Riemannian density √ g dt of the metric ds 2 = gdt 2 , the symmetry group is again Diff(I).
Let us now turn to the lax BV-BFV formulation of Jacobi theory. We first need to introduce the ghost field, which in case of diffeomorphims invariance is chosen to beξ∂ t ∈ X(I)[1] [Pig00]. In this setting, the Chevalley-Eilenberg operator acts on the fields as the Lie derivative γ J = Lξ ∂t and on the ghost as the Lie bracket of vector fields (cf. Example 2.2.5). We work with the componentξ ∈ C ∞ (I, R)[1] for simplicity.
Proof. It is a matter of a straightforward calculation to check that the formulas above satisfy the axioms of Definition 2.5.5.
Remark 3.5.3. Note that we can explicitly decompose the cohomological vector field Q J into its Chevalley-Eilenberg and Koszul-Tate parts as Q J = γ J + δ J by using Equation (4) and setting γ J = Q J − δ J on {q + ,ξ + }. We have: Asq is a function andξ is the component of a vector field, definingq + andξ + through Equation (22) lets us interpret them as components of tensor fields in and Ω top (I) ⊗ Ω top (I)[−2] respectively, or rather as components of a rank-(0, 1) and a rank-(0, 2) tensors over I. As such we see that the Chevalley-Eilenberg differential also acts as γ J = Lξ ∂t on {q + ,ξ + }.
In the case of 1D GR we have Proposition 3.5.4. The data , which are given by and the cohomological vector field Q GR ∈ X evo (F lax GR ) defines a lax BV-BFV theory.
Proof. It is straightforward to show that these formulas satisfy the axioms of Definition 2.5.5.
Before presenting the main theorem of this section we need to introduce some useful notation. Let v ∈ C ∞ (I, R n ) be a R n -valued field and let u =q/ q denote the normalised velocity of q. Note that u is always well-defined because we assumė q = 0. We can then decompose v = v + v ⊥ into its parallel v and perpendicular v ⊥ components with respect to u as where we used that T = m 2 q 2 . We now present the main theorem of this section, together with an outline of the proof. The computational details and the various required Lemmata are presented in the Appendix B.
Theorem 3.5.6. The lax BV-BFV theories F lax GR and F lax J of 1D GR and Jacobi theory are lax BV-BFV equivalent.
Proof. We need to check all the conditions from Definition 2.6.3. The existence of two maps φ, ψ with the desired properties is presented in Lemmata B.2.1 and B.2.2 respectively, where we also show that the pullback maps φ * , ψ * are chain maps w.r.t. the BV-BFV complexes BV-BFV • i , and that they map and maps the lax BV-BFV data of 1D GR as where η := gE T , and maps the lax BV-BFV data of the Jacobi theory as in accordance with our notion of lax BV-BFV equivalence. Furthermore, we need to show that the respective BV-BFV complexes are quasiisormophic. The composition map λ * = φ * • ψ * is shown to be the identity in Lemma B.2.3. In Lemma B.2.5, we prove that the composition map χ * = ψ * • φ * , which has explicitly form χ * ξ + = η 3/2 ξ + + g 3/2 Eġ + g + , is homotopic to the identity by constructing the morphism χ * s = e sLD GR with D GR = [R GR , Q GR ], where R GR is chosen to act as The homotopy is given by where σ(ϕ) = ϕ∂ t (g 3/2 g + ) −φg 3/2 g + , and fulfils lim s→∞ χ * s = χ * . There are some steps that are important to highlight in this case. First, in Lemma B.2.4 we show that R GR commutes with the Chevalley-Eilenberg differential γ GR

23
[R GR , γ GR ] = 0, by using general arguments and γ GR ∼ L ξ∂t , but it can also be checked through straightforward calculations. We do this as it greatly simplifies the computations Furthermore, while the computations for the action of χ * s = e sLD GR on the fields ϕ ∈ {q, ξ, g} is analogous to the ones presented for the other examples, it turns out that in this case of the antifields ϕ + ∈ {q + , ξ + , g + } finding a recursive formula for D k GR ϕ + is quite challenging. Instead, it is easier to take a slight detour: we first compute χ * s (g 3/2 ϕ + ) through D k GR (g 3/2 ϕ + ) and then use the property that χ * s = e sLD GR is a morphism in order to recover χ * s ϕ + χ * s (g 3/2 ϕ + ) = (χ * s g) The limit s → ∞ then reads Since χ * s g is nowhere vanishing for any s ∈ R ≥0 , this expression is well-defined ∀s ∈ R ≥0 iff χ * s (g 3/2 ϕ + ) is well-defined ∀s ∈ R ≥0 as well. We exemplify this procedure with the computation of χ * s g + . In order to see where the aforementioned problem arises we compute One can then proceed with the calculation of D k GR g + for higher k's and notice that the expressions become quite lengthy as D GR g = T /E − g (Equation (43)). The idea to avoid this complication by considering D k GR (g 3/2 g + ), where a recursive formula becomes apparent. We have It is then straightforward to see that D k GR (g 3/2 g + ) = (−1) k g 3/2 g + for k ≥ 0, ⇒ e sLD GR (g 3/2 g + ) = e −s g 3/2 g + .
Using Eq. (24) we then have The computations for q + , ξ + are lengthier and can be found in the Appendix B.2. In Lemma B.2.6 we demonstrate that χ * is the identity in cohomology by showing that the map therefore proving that the two lax BV-BFV theories in question have isomorphic BV-BFV cohomologies and thus that they are lax BV-BFV equivalent.
In this example, we are also interested on how the the composition maps λ * , χ * affect the boundary structure, namely the strict BV-BFV structure of the Jacobi theory and 1D GR. More specifically, we want to investigate how they change the kernel of the pre-boundary formsω and as such the quotient F ∂ =F ∂ / kerω (cf. Equation (12)).
In the case of λ * this is trivial since it is the identity. Regarding χ * , we argue that ker χ * ω GR has a singular behaviour and that we cannot construct a BV-BFV theory from the data χ * F lax GR := (F lax GR , χ * θ • GR , χ * L • GR , Q GR ). Thus, although χ * is the identity in the BV-BFV cohomology H • (BV-BFV GR ), it spoils the BV-BFV structure of 1D GR.
2.3.1. This is compatible with the process of removal of auxiliary fields outlined in [BBH95]. However, the request that two lax-equivalent theories both admit a strictification in the sense of Remark 2.5.6 is a genuine refinement of the notion of BV (and lax) equivalence of field theories. Since the BV-BFV quantization program requires a strict theory, this obstruction marks a roadblock for non-strictifiable lax BV-BFV theories.
Its pullback map φ * is a chain map w.r.t. (L Qi − d) and maps the lax BV-BFV data of the first-order theory as Proof. The computations are in the same line as the ones presented in the example of Classical Mechanics on a curved background (cf. Lemma 3.3.5). One should keep in mind that Q 2 ⋆FÃ = [c, ⋆FÃ].
Proof. Keeping in mind that φ * B † = 0, this is a straightforward calculation.
acts as and is homotopic to the identity.
Proof. The explicit computation for χ * is again straightforward. To show that it is indeed homotopic to the identity, we choose the vector field R 1 ∈ X evo (F lax 1Y M )[−1] to act as We now want to compute χ * s = e sLD 1 , with D 1 = [R 1 , Q 1 ], and show that lim s→∞ χ * s = χ * . Computation for A and c: Computation for B: Noting that D 1 A = 0 implies D 1 F A = 0, we have Computation for B † : Computation for A † : Computation for c † : Thus we have shown that χ * is homotopic to the identity.
Lemma A.1.5. The map χ * is the identity in cohomology.
Proof. We have to show that the map h χ converges on F lax 1Y M , namely As In the case of B we compute As such h χ converges and χ * is the identity in cohomology.
Appendix B. Lengthy Calculations for the Jacobi theory/1D GR case B.1. Preliminaries for calculations -tensor number. This appendix has two purposes. It serves as a preliminary for the computations, by presenting a straightforward way to compute the action of the Chevalley-Eilenberg differentials γ J , γ GR , and it provides an explaination for why they act as L ξ∂t on the antifields and antighosts (see Remarks 3.5.3 and 3.5.5). We will be using the 1D GR theory in this discussion but all considerations hold for the Jacobi theory as well. Let M be a manifold of arbitrary dimension and X = X σ ∂ σ ∈ X(M ). Recall that the Lie derivative L X acts on the components of a tensor field A ∈ T n m (M ) of rank-(n, m) as Let now M = I ⊂ R denote an interval and X = ξ∂ t ∈ X(I)[I] be the ghost field. In this setting Equation (26) is greatly simplified since µ i = ν i = t, where t is the coordinate on I. Let A ∈ T n m (I) and denote its component by A. We define the tensor number as t(A) = (m − n). We then have L ξ∂t A = ξ∂ t A − n∂ t ξA + m∂ t ξA = ξȦ + t(A)ξA.
As an example we list the tensor number for the fields, ghosts, antifields and antighosts of the 1D GR theory which explains why we claimed that the Chevalley-Eilenberg differential γ GR acts as L ξ∂t on the antifields and antighosts in Remark 3.5.5. As such we have γ GR = L ξ∂t on all the functions on {q, g, q + , g + , ξ + } and γ GR = 1 2 L ξ∂t on the ghost. Since the ghost is a special case, we assume that the tensor fields only depend on {q, g, q + , g + , ξ + } for the rest of the discussion. When computing γ GR (·), we then consider the parts with ghosts and without separately.
The discussion until now only holds for tensor fields that only depend on the 0th-jets of {q, g, q + , g + , ξ + }. The action of γ GR is then naturally extended to all jets since we assume that Q GR , and as such γ GR , is evolutionary, i.e. [L γGR , d] = 0. For example, if A only depends on 0th-jets then For a general tensor field A which depends on arbitrary jets of the fields we have for some real scalars t n (A) and some functions a n that depend on the jets of Φ, Φ + ∈ F GR . In order to extend the notion of tensor number to such objects we define Definition B.1.1. Let A be a tensor field that depends on arbitrary jets of Φ, Φ + ∈ F GR . The tensor number t(A) of A is defined as the scalar t 1 (A) in Equation (29).
Note that in order to compute γ GR we only have to find out what the t n (A) are. For most of the computations we are only going to encounter tensor fields that depend on the 0-th jets, and they will atmost include 2nd-jets. As such we want to find a pragmatic way of computing t(A). If necessary, we then look at higher t n (A), for example by following Equation (28). We list some useful properties of t(·), since they immensely simplify the explicit computations of γ GR (A).
Proposition B.1.2. Let A, B be two tensor fields that depend on an arbitrary number of jets of Φ, Φ + ∈ F GR . The tensor number has the following properties: (1) t(AB) = t(A) + t(B), Proof. Note that the only two terms from Equation (29) that can contribute to these properties are the first two. Therefore we will only show the computations for two tensor fields A, B that only depend on the 0th-jets, but they extend to the general case in a straightforward way. (2) Using that γ GR is a derivative we see that γ GR A n = nA n−1 γ GR A = nA n−1 [ξȦ + t(A)ξA] = ξ∂ t A n + nt(A)ξA n .
We finish this section by presenting the action of γ GR tensor numbers for some relevant quantities t(q) = 1 + t(q) = 1, We exemplify this method of calculating γ GR (A) with the computation of A = T = m 2 q 2 . Recall that γ GR q = ξq and as such γ GRq = ξq +ξq. γ GR T could potentially have terms proportional toξ since it depends on the derivativeq, but as there are no such terms in γ GRq there won't be any in γ GR T . As such we have B.2. Lemmata used in Theorem 3.5.6. In this subsection we explicitly present the lemmata used in Theorem 3.5.6 and the detailed calculations.
Its pullback map φ * is a chain map w.r.t. (L Qi − d) and maps the lax BV-BFV data of the first-order theory as The proof for the chain map condition φ * • Q GR = Q J • φ * is a matter of straightforward computations where η := gE/T and theq + ,q + ⊥ notation works as in Equation 23. Its pullback map ψ * is a chain map w.r.t. (L Qi − d) and maps the lax BV-BFV data of the first-order theory as Proof. We start with the chain map condition ψ * • Q J = Q GR • ψ * . In the case of the fields {q,ξ} we simply compute When dealing with the antifields {q + ,ξ + } it is useful to first show that ψ * is a chain map w.r.t. to the Chevalley-Eilenberg differentials and then proceed to show that is also fulfills this condition w.r.t. the Koszul-Tate differentials.
In the case of the Chevalley-Eilenberg differentials γ J , γ GR it is sufficient to investigate how ψ * changes the tensorial properties of the fields, i.e. to analyse the tensor number introduced Section B.1. Indeed using ψ * ξ = ξ we can compute The most general form of the other side of the chain map condition γ GR ψ * Φ+ is given by where the field dependent coefficients a n do not vanish trivially since the expressions for ψ * Φ+ depend on derivative terms such asġ,ġ + , andĖ L g . In order to show that the two sides of the chain map condition given in Equations (31) and (32) are equal we need prove that ψ * preserves the tensor number t(·) and that the coefficients a n vanish t(Φ + ) = t(ψ * Φ+ ), a n = 0.
The last term vanishes due to Equation (34). In order to show that the term proportional to g + vanishes as well we note: T Ω 2 Ṫ .
finishing the proof.
and as such the identity in cohomology.
We now prove that R GR commutes with the Chevalley-Eilenberg differential γ GR if R GR ξ = 0. Effectively, this means that we can ignore the Chevalley-Eilenberg part of Q GR in D GR = [Q GR , R GR ] and only have to regard the Koszul-Tate differential when explicitly computing the action of D GR . Recall that the Chevalley-Eilenberg differential acts as γ GR = L ξ∂t on {q, g, q + , g + , ξ + } and as γ GR = 1 2 L ξ∂t on {ξ}.
Lemma B.2.4. Let R GR ∈ X evo (F GR ) be an evolutionary vector field on F GR with the following properties • R GR vanishes on X[1](I), • R GR preserves the tensor rank on I, and let γ GR be the Chevalley-Eilenberg differential of the 1D GR theory. Then [R GR , γ GR ] = 0.
Proof. Recall that all the fields we are considering are components of tensor fields over I. In particular, note that the property that R GR vanishes on X[1](I) implies R GR ξ = 0, since ξ∂ t ∈ X[1](I). As γ GR ∼ L ξ∂t , it suffices to show that [R GR , L ξ∂t ] = 0 on functions, 1-forms and vector fields over I. We assume that all objects have an internal grading throughout the proof in order to account for the ghost number. where we used that R GR is evolutionary. As such it is sufficient to show that Ω • (I) ⊂ ker[R GR , ι ξ∂t ]. By definition all functions on I are in the kernel of ι ξ∂t : C ∞ (I) = Ω 0 (I) ⊂ ker ι ξ∂t . Let f ∈ Ω 0 (I). Since we assume that R GR preserves the tensor rank we also have R GR f ∈ Ω 0 (I), then R GR ι ξ∂t f = 0, ι ξ∂t R GR f = 0, and as such [R GR , ι ξ∂t ]f = 0. Let now ̟ = f dt ∈ Ω 1 (I) be a 1-form. Taking into account that |dt| = 1, we have where we used f ∈ Ω 0 (I) ⊂ ker[R GR , ι ξ∂t ], df ∈ Ω 1 (I) ⊂ ker[R GR , ι ξ∂t ] and d̟ = 0, since Ω 1 (I) = Ω top (I).
The strategy in the case of q + is the same as for g + and ξ + . Due to we have D GR (g 3/2 q + ) = D GR g 3/2 q + · mq 2T = D GR (g 3/2 q + ·q) mq 2T .
Where on the third line we used the integral in I ⊥ and similarly for the other integrals. Gathering everything results in h χ q + = (1 − η 3/2 ) ξ + + g 3/2 Eġ .