BV-BFV APPROACH TO GENERAL RELATIVITY ... - w w w .math.uzh.ch

5 downloads 0 Views 537KB Size Report
General Relativity (GR) in the Einstein Hilbert formalism (EH) is not manifestly .... extended to the level of ghost fields and antifields, hints at a strong regularity of.
BV-BFV APPROACH TO GENERAL RELATIVITY EINSTEIN-HILBERT ACTION A. S. CATTANEO AND M. SCHIAVINA

Abstract. The present paper shows that General Relativity in the ADM formalism admits a BV-BFV formulation in the sense of [1]. More precisely, for any d + 1 6= 2 (pseudo-)Riemannian manifold M with spacelike or timelike boundary components, the BV theory on the bulk induces a compatible BFV theory on the boundary. As a byproduct, the usual canonical formulation of General Relativity is recovered in a straightforward way.

Contents Introduction Acknowledgements 1. Classical BV and BFV formalisms 1.1. BV-BFV formalism for gauge theories 2. General Relativity in the BV formalism 2.1. ADM decomposition and boundary structure 2.2. Classical boundary structure 2.3. BV-BFV ADM theory 2.4. Constraint algebra and boundary gauge symmetry References

1 3 3 4 6 7 7 9 17 18

Introduction General Relativity (GR) in the Einstein Hilbert formalism (EH) is not manifestly a gauge theory, in the sense that it is not a theory of principal connections on some space-time manifold. It is nevertheless a field theory that admits a large group of transformations under which its action is invariant, namely the group of space-time diffeomorphisms. Generalised notions of gauge theories have been considered in the past where the requirement that the symmetry distribution be involutive has been relaxed to accommodate for more general cases like the Poisson sigma model or BF theories. The natural setting for a consistent treatment of such symmetries in view of (perturbative) quantisation is given by the Batalin-(Fradkin)-Vilkovisky [2, 3, 4] (B(F)V) formalism, that generalises the well known and celebrated Becchi Rouet Stora, Tyutin [5] (BRST) approach to the quantisation of gauge theories. The BRST and B(F)V formalisms provide a cohomological description of gauge equivalent classes of fields and of coisotropic submanifolds [6] (e.g. the critical locus of the action). The BFV formalism for the boundary can in principle be defined just in terms of the induced constraints and, under certain assumptions, This research was (partly) supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation, and by the COST Action MP1405 QSPACE, supported by COST (European Cooperation in Science and Technology). The authors acknowledge partial support of SNF Grants No. 200020-149150/1 and No. PDFMP2_137103. M.S. also acknowledges the Forschungskredit of the University of Zürich. 1

2

A. S. CATTANEO AND M. SCHIAVINA

it can be quantized to produce a complex whose cohomology in degree zero is the quantization of the reduced phase space. In the work of Cattaneo, Mnëv and Reshetikhin (CMR) [1, 7, 8] it has been shown how it is possible, under certain assumptions, to induce a compatible BFV structure on the boundary ∂M starting from a BV structure on the bulk space-time manifold M . The observation in CMR is that one needs a compatibility between the bulk BV structure and the boundary BFV structure for the state associated to the bulk to be a cocycle (and hence to define a physical state). At the semiclassical level, the fundamental compatibility condition is that the failure of the bulk action to be the Hamiltonian function for the BV operator should be given by pullback of the boundary Noether 1-form. This can always be achieved in terms of a larger space of boundary fields on which the differential of the Noether 1-form is degenerate. The crucial assumption is that the symplectic reduction of this 2-form should be smooth. In this paper we show that this is satisfied in the case of the EinsteinHilbert formulation of General Relativity. This has many advantages, such as a compatible cutting-gluing procedure, which allows to break topologically nontrivial manifolds into pieces, for which one might expect the quantisation to be simpler, as well as a powerful understanding of the quantisation procedure itself as a suitable generalisation of the Atiyah-Segal axioms for (topological) gauge theories [8, 9, 10]. Moreover, this approach gives a clear handle on the classical theory, providing a much cleaner understanding of the Hamiltonian approach to classical gauge theories, after Dirac and his canonical constraint analysis [11] which is in fact a first step towards their quantisation. As a matter of fact, we will work out here the necessary conditions of the aforementioned BV-BFV approach to General Relativity, i.e. the starting point for the program of BV quantum gravity. Even though most of the results that we will present are chiefly a recovery of established knowledge (above all [12] and subsequent citing literature), their straightforward derivation using the BV-BFV machinery is nontrivial and novel. Using a much weaker requirement than global hyperbolicity of the space-time manifold M we are able to prove that the boundary theory is the symplectic reduction of the canonical Hamiltonian formalism of general relativity in the ADM formalism [13], even when symmetries are dynamically taken into account. The induced data, encoded in a boundary action, will encompass at the same time all the relevant Hamiltonian and momentum constraints, together with the structure of the residual gauge transformations on the boundary. As a byproduct, we are able to address a recent question posed in [14], about the origin and nature of the constraint algebra, showing that it yields a nontrivial, non constant and yet linear coisotropic structure, not manifestly coming from a Lie algebra action. Indeed, the very compatibility of a BV structure in the bulk manifold with a BFV structure in the boundary means that the induced graded manifold given by the space of boundary fields automatically represents the cohomological resolution of the phase space of the theory. In Section 1 we will briefly review the ideas behind the BV-BFV formalism and the axioms one would like the gauge theory to satisfy in order to have access to the BV-BFV quantisation machinery [8]. Further on we will show (Theorem 2.1) that General Relativity in the Einstein Hilbert formalism does indeed satisfy the axioms, for all pseudo-Riemannian manifolds of dimension d+1 6= 2 with space/time-like boundaries, and for all Riemannian manifolds of dimension d 6= 2. Notice that our result does not rule out the presence of null or type-changing boundaries, but the technique has to be adapted, and it will be subject of further research.

BV-BFV GR: EINSTEIN HILBERT ACTION

3

In Theorem 2.2 we give explicit expressions in Darboux coordinates of the (0)symplectic graded space of boundary fields. Its degree 0 part is the well known symplectic space underlying the usual canonical description of Hamiltonian ADM GR, of which an inedited explicit expression we give in Section 2.2. Again, the classical canonical analysis is strongly simplified when using the present approach, which substantially stems from the interpretation of the Noether 1-form as a differential form on the space of boundary fields. The critical dimension d = 1 is different from the outset, the Einstein equations are trivial and the group of conformal transformations of the metric has to be considered explicitly. It is interesting to notice that the procedure that we use to construct the boundary structure will be able to single out the critical dimension. The explicit expressions for the boundary data will indeed be singular in d = 1, hinting at a different behaviour. This is the first of a series of papers dealing with the BV-BFV approach to quantum gravity. In a subsequent work [15] we will show a similar analysis of the Palatini-Holst formalism of GR, highlighting interesting similarities and differences. In [16] we will perform the actual boundary BV quantisation of one dimensional examples of pure gravity and cosmological models. There we will show how time evolution can be recovered by means of an apparently silly breaking of diffeomorphism invariance, and how this can be interpreted as a natural operation. Lastly, we would like to stress that this bulk-boundary correspondence, when extended to the level of ghost fields and antifields, hints at a strong regularity of the theory in terms of compatibility with the gauge symmetries. This is, possibly, to be taken as a necessary requirement for a theory to be regarded as neatly quantisable, or at least as a mean of distinction between classical theories, otherwise indistinguishable. Acknowledgements The authors would like to thank J. Stasheff for his helpful comments, and P. Mnëv for relevant remarks on earlier stages of the work. 1. Classical BV and BFV formalisms We will consider here a general framework for gauge field theories. First of all we fix the space dimension, say d, and assign to a d-dimensional manifold M (possibly with boundary, and other geometric data, like a Riemannian structure) a space of fields FM , i.e. a Z-graded odd-symplectic manifold, with a symplectic form ΩM of degree |ΩM | = k together with a local, degree k + 1 functional SM of the fields and a finite number of their derivatives. The equations of motion (i.e. the dynamical content of the theory) are encoded in the Euler Lagrange variational problem for the functional SM . The Z grading is sometimes called ghost number, but it will be often replaced by the computationally friendly total degree, which takes into account the sum of different gradings when the fields belong to some graded vector space themselves (e.g. differential forms). The symmetries are encoded by an odd vector field QM ∈ T [1]M such that [Q, Q] = 0. A vector field with such a property is said to be cohomological. Among these pieces of data some compatibility conditions are required. We give the following definitions for different values of k. Our model for a bulk theory will be given by Definition 1.1. A BV-theory on a closed manifold M is the collection of data (FM , SM , QM , ΩM ) with (FM , ΩM ) a Z-graded (−1)-symplectic manifold, and SM and QM respectively a degree 0 function and a degree 1 vector field on FM such that

4

A. S. CATTANEO AND M. SCHIAVINA

(1) ιQM ΩM = δSM , i.e. SM is the Hamiltonian function of QM (2) [Q, Q] = 0, i.e. QM is cohomological. The symplectic structure defines an odd-Poisson bracket (, ) on FM and the above conditions together imply (S, S) = 0 (1) the Classical Master Equation (CME). On the other hand, the model for a boundary theory, induced in some sense to be explained, will be given by Definition 1.2. A BFV-theory on a closed manifold N is the collection of data (FN , SN , QN , ΩN ) with (FN , ΩN ) a Z-graded 0-symplectic manifold, and SN and QN respectively a degree 1 function and a degree 1 vector field on FN such that (1) ιQN ΩN = δSN , i.e. SN is the Hamiltonian function of QN (2) [Q, Q] = 0, i.e. QN is cohomological. This implies that SN satisfies the CME. In general one starts from a classical theory, that is an action functional Scl for some space of classical fields FM and a distribution DM in the bulk encoding the symmetries, i.e. LX (Scl ) = 0 for all X ∈ DM . The main requirement on DM for the formalism to make sense is that DM be involutive on the critical locus of Scl . Notice that DM can be the distribution induced by a Lie algebra (group) action, in which case it is involutive on the whole space of fields. When this is the case we will talk of BRST formalism, even though the setting will be slightly different from the original one (for another account on the relationship between the BV and BRST formalism see, e.g. [17]). We will be mainly interested in these types of theories, but for the sake of completeness we will sketch the general construction. To construct a BV theory on the bulk starting from classical data, and assuming that M has no boundary, we must first extend the space of fields to accommodate for the symmetries: FM FM = T ∗ [−1]DM [1]. Symmetries are considered with a degree shift of +1, whereas the dualisation introduces a different class of fields (called anti-fields) with opposite parity to their conjugate fields, owing to the −1 shift in the cotangent functor. This yields a (−1)-symplectic manifold, which is a good candidate to be the space of bulk fields we want to work with. The classical action has to be extended as well to a new local functional on FM , and if we want this to satisfy the axioms of the BV theory we must impose the CME on the extended action. This process will a priori need the introduction of higher degree fields to the space of fields in order to resolve, under some regularity assumptions, the relations among degree 1 fields. This process of extension goes through co-homological perturbation theory [6, 19, 2, 18, 7] and it will ensure us to end up with a BV structure on the bulk. However, for a theory which is BRST-like, the extension is determined by the following straightforward result [2]: Theorem 1.1. If DM comes from a Lie algebra action, the functional SBV = Scl +hΦ† , QM Φi on the space of fields FM = T ∗ [−1]DM [1] satisfies the CME, where Φ is a multiplet of fields in DM [1], and QM is the degree 1 vector field encoding the symmetries of DM . FM is then a (−1)-symplectic manifold and together with SBV and QM it yields a BV theory that (minimally) extends the classical theory. 1.1. BV-BFV formalism for gauge theories. As we already mentioned, Definition 1.2 will be a boundary model for Definition 1.1. In what follows we will explain in which sense. Say that we start from the data defining a BV theory, but this time we allow M to have a boundary: the requirement that ιQM ΩM = δSM

BV-BFV GR: EINSTEIN HILBERT ACTION

5

is (in general) no longer true. What will happen is that the integration by parts one usually has to take into account when computing δS will leave some non zero terms on the boundary. More precisely, consider the map π e : FM −→ Fe∂M (2) that takes all fields to their restrictions to the boundary and their jets (it is a surjective submersion). We can interpret the boundary terms as the pullback of a one form1 α e on Fe∂M , namely ιQM ΩM = δSM + π e∗ α e

(3)

We will call α e the pre-boundary one form. Notice that if we are given this data, we can interpret this as a broken BV theory, which induces some data on the boundary. We can in fact consider the preboundary two form ω e := δ α e and if it is pre-symplectic (i.e. its kernel has constant ∂ rank) then we can define the true space of boundary fields F∂M to be the symplectic reduction of the space of pre-boundary fields, namely:  ∂ F∂M = FeM ker(e (4) ω) ∂ with projection to the quotient denoted by π : FeM −→ F∂M . If all of the above assumptions are satisfied, the map πM := π ◦ π e is a surjective submersion, the ∂ := ω reduced two form ω∂M e is a 0-symplectic form, and we have the following

Proposition 1.1 (CMR [7]). The cohomological vector field QM projects to a coho∂ mological vector field Q∂∂M on the space of boundary fields F∂M . Moreover Q∂∂M is ∂ Hamiltonian for a function S∂M , the boundary action. We can now take this as a definition, as follows Definition 1.3 (CMR [1]). A pre-BV-BFV theory on a d-dimensional manifold M with boundary ∂M is a collection of data (FM , SM , QM , ΩM ) with (FM , ΩM ) a Zgraded (−1)-symplectic manifold, and SM and QM respectively a degree 0 function and a degree 1 vector field on FM such that (1) [QM , QM ] = 0, i.e. QM is cohomological, (2) The map π e to the space of pre-boundary fields Fe∂M is a surjective submersion. e on Fe∂M (3) QM is π e-projectable to a cohomological vector field Q (4) The BV-BFV formula ιQ ΩM = δSM + π e∗ α e is satisfied. Whenever the pre-boundary 2-form ω e is pre-symplectic on FeM and the symplectic ∂ ∂ reduction to the space of boundary fields (F∂M , ω∂M ) can be performed, this induces ∂ ∂ ∂ ∂ the BFV theory (F∂M , S∂M , Q∂M , ω∂M ) on ∂M . ∂ The composition of π e with the symplectic reduction map π : FeM −→ F∂M will ∂ yield another pre-BV-BFV theory, for the symplectic form ω∂M and the surjective ∂ submersion πM = π ◦ π e : FM −→ F∂M satisfying axioms from (1) to (3). In this ∂ case we say that the theory is BV-BFV. Furthermore, if α e is basic, α e = π ∗ α∂M , we say that the BV-BFV theory is exact and we have the fundamental formula ∗ ∂ ιQM ΩM = δSM + πM α∂M

(5)

The advantage of such a point of view is at least twofold. First of all, as we just saw, the formalism is large enough to be able to describe consistently what happens both in the bulk and in the boundary. On the other hand it is flexible enough to allow for symmetries that are more general than a Lie group action. 1In full generality α e is a connection on a line bundle, yet when SM is a function on the space of fields, α e is a globally well defined 1-form.

6

A. S. CATTANEO AND M. SCHIAVINA

For instance it is possible to accomodate symmetries that close only on shell (e.g. Poisson sigma model) or symmetries whose generators are not linearly independent, where higher relations among the relations are required (e.g. BF theory or other theories involving (d > 1)-differential forms. The BV theory that we have constructed in Theorem 1.1 when a gauge theory of the BRST-kind was given is sometimes called the minimal BV extension of the gauge theory. When a non trivial boundary is allowed, we will anyway use this minimal extension as the starting point for the BV-BFV analysis. What one aims to establish is whether this minimal BV theory on the bulk is indeed a BV-BFV theory. We will accomplish this for the special case of General Relativity in the EH-ADM formalism in what follows. 2. General Relativity in the BV formalism General Relativity (GR) is a classical theory modelling the gravitational interaction between physical objects. It can be expressed as the variational problem for a functional on some space of classical fields. The usual presentation of GR in the second order formalism requires one to consider the basic field to be a pseudoRiemannian metric g on a spacetime manifold M , i.e. a d + 1 dimensional manifold, possibly with non trivial boundary ∂M , and the so-called Einstein Hilbert action (up to constants) Z √ EH (6) Scl = (R[g] − 2Λ) −g dd+1 x M

where R[g] is the Ricci scalar of the pseudo-Riemannian metric g ∈ PR(d,1) (M ) with signature (d, 1), g := det(g) and Λ is the cosmological constant. The Einstein equations, which determine the dynamics of g, are derived from the action as Euler Lagrange equations for the variational problem. The very principle of general covariance, from which the whole of General Relativity stems, states that the theory should be the same in all reference frames or, equivalently, that the equations of motion should not depend on the choice of coordinates, and this is mathematically encoded in the action being invariant under the full group of diffeomorphism of the manifold M . To make contact with the general theory outlined in Section 1 we will consider the distribution defining the (infinitesimal) gauge symmetries D to be the maximal distribution given by all vector fields on M . This will be encoded in the BV formalism by the introduction of a new field ξ ∈ Γ (T [1]M ), which is an odd vector field representing such infinitesimal symmetries, and the space of fields will be enlarged accordingly, as we shall see in detail later on. It turns out that a clean result can be obtained, when some slightly restricting conditions are imposed on the boundary ∂M : namely, we will require that ∂M be either space-like or time-like, and this will allow us a clever rewriting of the action in a boundary-friendly fashion. To achieve this in field theory, since the pseudo-Riemannian structure is not fixed, we will need to restrict the space of fields to only those metrics whose restriction to the boundary has either space-like or time-like signature. Such a space of fields will be denoted by PR∂M (d,1) (M ). Observe that in the literature (e.g. [13, 12]), it is customary to require that the spacetime manifold M be globally hyperbolic, or that it has the product structure Σ×R for Σ an embedded space/time-like submanifold of M . This would indeed be a much stronger requirement, and in fact we only require it be true in a neighborhood of the boundary. Notice that any globally hyperbolic Lorentzian structure for the space-time M is in particular contained in PR∂M (d,1) (M ).

BV-BFV GR: EINSTEIN HILBERT ACTION

7

2.1. ADM decomposition and boundary structure. Among the possible boundaries that one can consider there is the class of non null boundary, described locally by a submanifold of the form xn = const. As we mentioned, this is equivalent to (or rather means) asking that the space of pseudo-Riemannian structures on the manifold with boundary M be limited to those metrics whose restriction to the boundary has either time-like or space-like signature. When this is the case, and when the xn component corresponds to a signature2 −, the metric and its inverse are re-written in the form:   −(η 2 − βa β a ) βb gµν =  βa γab (7)   −1 βb µν −2 g = η β a η 2 γ ab − β a β b p √ With this decomposition we have that −g = η |γ| , with γ = detγij , and |γ| means that we shall consider the absolute value of the determinant if needed (if √  = −1). We will understand this fact from now on and simply write γ. The classical Einstein Hilbert action gets rewritten as Z n √ S= η γ((Kab K ab − K 2 ) + R∂ − 2Λ) + {z } | M (8) LADM o √ √ √ ab a d+1 − 2∂n ( γK) + 2∂a ( γKβ − γγ ∂b η) d x where we define Kab , the second fundamental form of the boundary submanifold, and its trace K by means of the boundary covariant derivative ∇∂ as follows 1 −1 1 η (2∇∂(a βb) − ∂n γab ) = η −1 Tab 2 2 1 1 K = γ ab Kab = η −1 γ ab Tab = η −1 T 2 2

Kab =

(9) (10)

while Tab and T are introduced for later convenience. The classical space of fields in this case is then simply given by Fcl = PR∂M (d,1) (M ) the space of pseudo Riemannian metrics on M with signaure (d,1), and space/time-like signature when restricted to the boundary. Remark 1. The total normal derivative appearing in (??) is the so called GibbonsHawking-York boundary term. In our framework, it will only affect the boundary 1form by an exact term, and will not interfere with the rest of the boundary structure. The action we will consider from now on is Z SADM = LADM (11) M

2.2. Classical boundary structure. To start off, we will consider first the classical (i.e. non-BV) structure that is induced on the boundary. This is often called canonical analysis, and one replaces the Lagrangian description with the Hamiltonian in the phase space of the system. The advantage in applying our variational approach to the classical case as well, is that we are able to perform the symplectic reduction of the space of classical pre-boundary, to find a well defined symplectic structure on the space of classical boundary fields, i.e. the phase space, encoding the canonical relations in a straightforward way. 2For simplicity  = 1 if xn is a timelike direction, that is to say when the boundary is spacelike.

8

A. S. CATTANEO AND M. SCHIAVINA

Proposition 2.1. The space of classical boundary fields for General Relativity in the ADM formalism for any dimension d + 1 6= 2 is an exact symplectic manifold. In a local chart the symplectic form reads Z ∂ ω = δγ ab δΠab (12) ∂M

where the projection map to the boundary fields reads: ( γ ij = γij   √ πM : γ Πlm = 2 Jelm − γlm γ ij Jeij

(13)

with Jelm = η −1 Jlm − 2∇(l βm)



(14)

and Jab is the first normal jet of γab evaluated at the boundary. Proof. Consider the variation of the action SADM : it splits in a bulk term and a boundary term. The latter is interpreted as a 1-form α e on the space of precl boundary fields Fe , which is given by restrictions of the bulk metric and its normal jets Jab := ∂n γab ∂M to ∂M :  Z  √ γ ab √ ab δγ Kab (15) α e = 2 δ( γγ )Kab − 2 ∂M

and the two form ω e = δα e, using the definitions (9) of Kab and Tab is ( Z √ √ ω e= δη −1 δ( γγ ab )Tab − η −1 δ( γγ ab )δTab + ∂M

−η

−1 δ



) √ √ γδγ ab −1 γ ab −1 γ ab Tab − δη δγ Tab + η δγ δTab (16) 2 2 2

Both K and T are functions of g and J. Observe that the transversal jets Jna are not present because of the clever rewriting of the action. After some straightforward calculations one is able to gather that the kernel of the two form is given, for d 6= 1 by (Xγ )ab = 0

(17)

(XT )lm = −η(Xη−1 )Tlm

(18)

but the (Xβm ) component of a vector field in the kernel turns out to be free, as well as the η −1 component. In fact, equation (18) can be unfolded to yield: (XJ )lm = −η(Xη−1 )Jlm + 2∇(l (Xβ )m) + 2η(Xη−1 )∇(l βm)

(19)

The generators in the kernel are δ δ δ − η(Xη−1 )Jlm + 2η(Xη−1 )∇(l βm) −1 δη δJlm δJlm δ δ Bl = (Xβ )l + 2∇(l (Xβ )m) δβl δJlm

E−1 = (Xη−1 )

(20a) (20b)

and thus, solving the differential equations given by the kernel vector fields (20) together with Equation (17) yields the symplectic reduction map: ( γ eij = γij  π: (21) Jelm = η −1 Jlm − 2∇(l βm)

BV-BFV GR: EINSTEIN HILBERT ACTION

It is a matter of a simple check to verify that the one form Z p   γ e ∂ δe γ ij γ eij γ elm Jelm − δe γ lm Jelm α = 2

9

(22)

∂M

pulls back along π to the pre-boundary one form α e: π ∗ α∂ = α e e = ιB l α e = 0. This implies that the symplectic which is horizontal,  i.e. ιE−1 α manifold F ∂ , δα∂ is exact. πM is then obtained composing π with the restriction to the pre-boundary fields π e.  √  γ Introducing the new variables γ ab ≡ γ eab and Πlm = 2 Jelm − γlm γ ij Jeij we have Z Z α∂ = −

δγ ab Πab =⇒ ω ∂ = 

∂M

δγ ab δΠab

(23)

∂M

which is the symplectic form in the space of classical boundary fields, expressed in local Darboux coordinates. X Remark 2. Using the classical Noether form for the classical ADM theory in the bulk we managed to recover the phase space description of General Relativity in the symplectic framework. Notice that in the non-BV setting the compatibility with ∗ ∂ the boundary structure is encoded in the boundary term πM α∂M , a failure of the variation of the action from being given by the Euler Lagrange equations alone. When turning to the BV theory we will see how this compatibility can be enriched to yield the full fundamental formula (5). Remark 3. Observe that we have performed a symplectic reduction that encodes the usual canonical analysis of General Relativity (in the ADM formalism). Our boundary field Πab is a projected version of the usual (i.e. literature) momentum coordinate conjugate to γ ab (let us call it pab = π ∗ Πab ), with the difference that in the present case the conjugacy is in the symplectic sense, as we quotient by the kernel of the pre-symplectic form ω e. In what follows we will show how this can be extended to the BV setting, which explicitly encodes the symmetries. This will allow us to recover the usual energy and momentum constraints in a straightforward way, still holding on to the clean symplectic description of the phase space. 2.3. BV-BFV ADM theory. Recalling the general theory we outlined in Section 1, in order to perform a consistent analysis of the theory including the symmetries, one has to find the correct BV data. The geometric information we need is the distribution in the space of fields that generates the symmetries. In our case, General relativity is symmetric under the action of the whole diffeomorphism group of the space-time manifold M. The theory can be treated as a BRST-like theory since the symmetry algebra T M closes everywhere in the space of fields, and we can use Theorem 1.1 to extend the classical ADM action to its BV-extended counterpart. Indeed we consider the following action: Z Z 1 BV ι[ξ,ξ] ξ † ≡ SADM + SBV (24) SADM = SADM − (Lξ g) g † + 2 M

M

with SADM as in (11), since the action of the cohomological vector field Q reads Qg = Lξ g 1 Qξ = [ξ, ξ] 2

(25)

10

A. S. CATTANEO AND M. SCHIAVINA

and ξ ∈ T [1]M a generic vector field, declared to have ghost number 1. The space of fields is then given by h i FADM := T ∗ [−1] PR∂M (M ) ⊕ Γ (T [1]M ) . (26) d+1 Our first result in this setting is the following BV Theorem 2.1. For all d 6= 1, the data (FADM , SADM , Q, ΩBV ) induce an exact BVBFV theory. The induced data on the boundary will be denoted by (F ∂ , S ∂ , Q∂ , ω ∂ ). In particular we have that Q∂ = πM ∗ Q, and ιQ∂ ω ∂ = δS ∂ . BV Proof. The variation of SADM induces the following pre-boundary 1-form on the e space of pre-boundary fields FADM , which is given by restrictions of the bulk fields to ∂M , together with the normal jets of the boundary metric Jab := ∂n γab ∂M :  Z  √ γ ab √ α eADM = 2 η δ( γγ ab )Kab − δγ Kab 2 Z ∂M  + 2 (−η 2 + βa β a )δξ n g †nn + βa δξ n g †an + βa δξ a g †nn + γab δξ (a g †b)n

Z ∂M Z  − ξ n δ(−η 2 + βa β a )g †nn + 2ξ n δβa g †an + ξ n δγab g †ab − ξ n δξ ρ χρ ∂M

(27)

∂M

and 2-form ω e = δα eADM :   Z √ ab −1 γ −1 √ ab ω e= δ η δ( γγ )Kab − η δγ Kab − δξ n δξ ρ χρ + ξ n δξ ρ δχρ 2 ∂M  +  δ(−η 2 + βa β a )δξ n g †nn + 2(−η 2 + βa β a )δξ n δg †nn + 2βa δξ n δg †an  + 2δβa δξ a g †nn + 2βa δξ a δg †nn + 2δγab δξ (a g †b)n + 2γab δξ (a δg †b)n   −  ξ n δ(−η 2 + βa β a )δg †nn + 2ξ n δβa δg †an + δξ n δγab g †ab + ξ n δγab δg †ab (28) where we fixed the volume form v = dxn ∧ v ∂ . Recalling that Kab is a function of Jab := ∂n γab |∂M , it is just a matter of lengthy computations to show that ω e is presymplectic: indeed, excluding the case d = 1, the equations defining the kernel can be solved to yield: (XJ )lm = + η −1 (Xη )Jlm + 2∇(l (Xβ )m) − 2η −1 (Xη )∇(l βm)   1 4 γlm βa − β(l γm)a g †an ξ n + √ (Xη ) γ d−1   1 4 γlm (Xβ )a − (Xβ )(l γm)a g †an ξ n −√ η γ d−1   2 1 + √ (Xη ) γlm γab − γla γbm g †ab ξ n γ d−1   2 1 −√ η γlm γab − γal γbm (Xg† )ab ξ n γ d−1

(29)

(Xg† )bn = − γ ab (Xβ )a g †nn + η −1 β b (Xη )g †nn + η −3 β b β a χa (Xη )ξ n (30)  −2 b a  − η β β (Xχ )a ξ n − η −2 β b (Xβ )c γ cd χd ξ n − η −3 β b (Xη )χn ξ n 2 2    + η −2 β b (Xχ )n ξ n − η −1 γ ba χa (Xη )ξ n + γ ba (Xχ )a ξ n 2 2 2

BV-BFV GR: EINSTEIN HILBERT ACTION

 (Xg† )nn = − η −1 (Xη )g †nn − η −3 (Xη )β a χa ξ n + η −2 β a (Xχ )a ξ n 2   + η −2 (Xβ )b γ ab χa ξ n + η −3 (Xη )χn ξ n − η −2 (Xχ )n ξ n 2 2

11

(31)

(Xξ )a = + β a η −1 (Xη )ξ n − γ ab (Xβ )b ξ n

(32)

(Xξ )n = − η −1 (Xη )ξ n

(33)

and the kernel is generated by the (vertical) vector fields:  δ  δ δ (34a) − η −2 (Xχ )n ξ n †nn + β b η −2 (Xχ )n ξ n †bn δχn 2 δg 2 δg δ  δ X(a) =(Xχ )a + η −2 β a (Xχ )a ξ n †nn (34b) δχa 2 δg   δ  − η −2 β b β a (Xχ )a ξ n − γ ba (Xχ )a ξ n 2 2 δg †bn δ  δ δ − γ ab (Xβ )a ξ n b + η −2 γ ab (Xβ )a χb ξ n †nn (34c) B(a) =(Xβ )a δβa δξ 2 δg     4 1 δ + 2∇(l (Xβ )m) + √ η (Xβ )(l γm)a − γlm (Xβ )a g †an ξ n γ d−1 δJlm    δ + − η −2 β b γ cd (Xβ )c χd ξ n − γ ab (Xβ )a g †nn 2 δg †bn   1 2 δ δ γlm γab (Xg† )ab ξ n G†(ab) =(Xg† )ab †ab + √ η γal γbm − (34d) δg γ d−1 δJlm δ δ δ (34e) E =(Xη ) − η −1 (Xη )ξ n n + β a η −1 (Xη )ξ n a δη δξ δξ δ δ − η −1 (Xη )g †nn †nn − η −3 (β a χa − χn ) (Xη )ξ n †nn δg δg    δ − η −3 β b χn − η −3 β b β a χa + η −1 γ ba χa (Xη )ξ n †bn 2 δg   4 1 δ − √ (Xη ) β(l γm)a − γlm βa g †an ξ n γ d−1 δJlm   1 2 δ − √ (Xη ) γla γbm − γlm γab g †ab ξ n γ d−1 δJab  δ δ + η −1 β a (Xη )g †nn †an + η −1 (Xη ) Jlm − 2∇(l βm) δg δJlm X(n) =(Xχ )n

It is easy to check that the boundary one form (27) is annihilated by all vertical vector fields (34), and it is therefore basic, proving the exacteness of the BV-BFV structure and concluding the proof. X Not only can we prove the existence of a well defined exact BFV structure on the boundary ∂M , but it is possible to express it in Darboux coordinates. The explicit expression in a local chart is established by the following

12

A. S. CATTANEO AND M. SCHIAVINA

∂ Theorem 2.2. The surjective submersion πM : FADM −→ FADM is given by the local expression: √    γ e   Jelm − γ elm γ eij Jeij Π = lm  2      = −2 ηg †nn − 2 η −1 (β a χa − χn ) ξ n ϕn    = 2 γab g †bn + γ ba βa g †nn − 2 γ ba χa ξ n πM : ϕa (35)   ξb = ξ b + γ ba βa ξ n      ξn = η ξn    γ ab = γab

with (

   1 2 γal γbm − γlm γab g †ab ξ n η −1 Jlm − 2∇(l βm) − √ γ d−1 )     4 1 2 1 †bn n b †nn n − √  β(l γm)b − γlm βb g ξ − √ β(l βm) − γlm βb β g ξ γ d−1 γ d−1

Jelm =

The boundary symplectic structure on the space of boundary fields in these coordinates (ρ = {n, a}) reads: Z ω∂ =  δγ ab δΠab + δξ ρ δϑρ . (36) ∂M

Moreover, the boundary action is given by the expression ) Z (  √    ab a n 2 ∂ ab ∂ Π Πab − Π + γ R − 2Λ + ∂a (ξ ϕn ) − γ ϕb ∂a ξ ξ n S = √ γ ∂M ) Z (  c cd cd + − ∂c γ Πda − (∂a γ )Πcd + ∂c (ξ ϕa ) ξ a . (37) ∂M

Proof. Using the vertical vector fields in (34) to eliminate βa , χρ and g †ab one is able ∂ to find an explicit section of the symplectic reduction π : FeADM −→ FADM . First 0 of all, use B to set βa = 0, this implies (Xβ )a = −βa together with βa (t) = (1 − t)βa0 and we have the first two differential equations: ξ˙b = +γ ab βa0 ξ n  g˙ †nn = − η −2 γ ab βa0 χb ξ n 2

(38a) (38b)

that are easily solved to yield ξ(t) =ξ0b + γ ba βa0 ξ n t  g †nn (t) =g0† nn − η −2 γ ab βa0 χb ξ n t 2 we use (39b) and the time rule for βa (t) to solve equation  g˙ †nb =+ η −2 β b γ cd βc0 χd ξ n + γ ab βa0 g †nn 2  =γ ba βa0 g0† nn − η −2 γ ba βa0 γ cd βc0 χd ξ n (2t − 1) 2  g †nb (t) =g0† nb + γ ba βa0 g0† nn t− η −2 γ ba βa0 γ cd βc0 χd ξ n (t2 − t) 2

(39a) (39b)

BV-BFV GR: EINSTEIN HILBERT ACTION

13

together with     4 1 0 0 0 †an n ˙ Jlm = − 2∇(l βm) + √ η β(l γm)a − γlm βa g ξ γ d−1   4 1 0 0 0 = − 2∇(l βm) − √ η β(l γm)a − γlm βa g0† an ξ n + γ d−1   1 2 0 0 0 b −√ β(l βm) − γlm βb β0 g0† nn ξ n t γ d−1   4 1 0 0 Jlm = − 2∇(l βm) t − √ η β(l γm)a − γlm βa0 g0† an ξ n t+ γ d−1   1 2 0 0 β(l βm) − γlm βb0 β0b g0† nn ξ n t2 −√ γ d−1 So we can set the temporary value of our fields at t = 1 to be   1 4 0 0 γm)a − γlm βa0 g0† an ξ n + Jˆlm = − 2∇(l βm) − √ η β(l γ d−1   2 1 0 0 β(l γlm βb0 β0b g0† nn ξ n −√ βm) − γ d−1 gˆ†nb =g †nb + γ ba βa g †nn  gˆ†nn =g †nn − η −2 γ ab βa χb ξ n 2

(40a) (40b) (40c)

Now we can turn to the vector fields Xρ and use them to set χρ = 0 at some value of the internal evolution parameter s. As usual we impose (Xχ )ρ = −χ0ρ and χρ (t) = (1 − t)χ0ρ . The new equations are  g˙ †nn =+ η −2 χ0n ξ n 2  g˙ †nb =− γ ba χ0a ξ n 2 which will yield an additional correction to the temporary value of our fields:   (41a) gˆ ˆ†nn =g †nn + η −2 χn − γ ab βa χb ξ n 2  gˆ ˆ†nb =g †nb + γ ba βa g †nn − γ ba χa ξ n (41b) 2 Similar is what happens when we use G†ab , for we get the equation   2 1 ˙ Jlm = − √ η γal γbm − γab γlm g0† ab ξ n γ d−1 that will correct the temporary value of Jˆlm to   4 1 ˆ 0 0 Jˆlm = − 2∇(l βm) − √ η β(l γm)a − γlm βa0 g0† an ξ n + γ d−1   2 1 0 0 0 b −√ β(l βm) − γlm βb β0 g0† nn ξ n γ d−1   2 1 − √ η γal γbm − γab γlm g0† ab ξ n γ d−1

(42)

Finally, we use the vector field E to set η = 1. This implies that the time law for η be given by η(t) = (1 − η0 )t + η0 and (Xη ) = 1 − η0 . The associated equations

14

A. S. CATTANEO AND M. SCHIAVINA

read 1 − η0 ξn (1 − η0 )t + η0 1 − η0 g †nn =− (1 − η0 )t + η0 1 − η0 = Jlm (1 − η0 )t + η0

ξ˙n = − g˙ †nn J˙lm

yielding, at time t = 1, the following corrections to the fields: ξen = ηξ n , ge†nn = ˆ η gˆˆ†nn and Jelm = η −1 Jˆlm . Putting everything together we get that the symplectic reduction map reads:     1  γlm γab g †ab ξ n Jelm = η −1 Jlm − 2∇(l βm) − √2γ γal γbm − d−1         4 1 2 1 †bn n b  √ √ − β γ − γ β g ξ − β β − γ β β g †nn ξ n  lm b lm b (l m)b (l m) γ d−1 γ d−1     ge†nn = ηg †nn + 2 η −1 (χn − β a χa ) ξ n π : ge†bn = g †bn + γ ba β g †nn +  γ ba χ ξ n a a  2   b ba n  eb ξ = ξ + γ β ξ  a    n en  ξ = ηξ    γ eab = γab (43) The boundary 1-form α∂ will be given by the ansatz ) Z (p   γ e ab lm e lm e n †nn a †bn ∂ δe γ γ eab γ e Jlm − δe γ Jlm − 2δ ξe ge + 2γab δ ξe ge (44) α = 2 ∂M ∗ ∂ as it is straightforward eADM . Introducing the new variables √ to check that πα = α γ e ab ab ij e e γ ≡γ e , Πab = Jab − γ eab γ e Jij together with ϑn = −2e g †nn , ϑa = 2e γab ge†bn 2

and ξ ρ = ξeρ , we can write the symplectic boundary form as: Z ω∂ =  δγ ab δΠab + δξ ρ δϑρ

(45)

∂M

and recover expression (35) and (37) for the projection and the boundary action in the Darboux coordinates. We would like to compute now the cohomological boundary vector field. First of all we must extract the analogue bulk vector field, encoding the equations of motion and the symmetries of the system, using the fundamental formula: ∗ ιQ ΩBV = δS + πM α∂

(46)

A shortcut to do this in the ADM formalism, instead of computing cumbersome integrations by parts, consists in considering the classical Einstein Hilbert action, whose classical vacuum equations of motion are given by     1 √ γ Rµν − R − Λ gµν ≡ Gµν = 0 2 and to express them using the ADM decomposition. This is done projecting the above equation on the new field direction, with the help of the Gauss-Codazzi equations and the Ricci equations. Doing so, one obtains the projection of the relevant Euler Lagrange terms in the ADM formalism, namely, forgetting about the BV extension for a moment:

BV-BFV GR: EINSTEIN HILBERT ACTION

  δSADM √ =  γ  R∂ − 2Λ + K 2 − Kab K ab δη   δS 1 √ √ := ADM = 2γ ba ∂c ( γγ cd Kda ) + ∂a γ cd Kcd − γ∂a K δβb 2  δS √ := ADM =  γ ∂n Kab − β k ∂k Kab − 2Kk(a ∂b) (g kc βc ) δγab

Gη := Gβa Gγab

15

(47) (48) (49)

Notice that the formula for Gβa is only apparently different from the usual momentum constraint that can be found in the literature (see e.g. [12]): Hc :=



γγ ba γ cd ∇∂c Kda − ∇∂a K



as it can be seen by manipulating the covariant derivatives. Adding the BV part we have that the derivatives of the action with respect to the new fields read:    δSBV = −2η ∂ρ ξ ρ g †nn + ξ ρ ∂ρ g †nn − 2∂ρ ξ n g †nρ δη    δSBV = 2 ∂ρ ξ ρ g †an + ξ ρ ∂ρ g †an − ∂ρ ξ a g †nρ − ∂ρ ξ n g †aρ + δβa  + 2β a ∂ρ ξ ρ g †nn + ξ ρ ∂ρ g †nn − 2∂ρ ξ n g †nρ     δSBV =  ∂ρ ξ ρ g †ab + ξ ρ ∂ρ g †ab − 2∂ρ ξ (a g †b)ρ + δγab  − 2β a β b ∂ρ ξ ρ g †nn + ξ ρ ∂ρ g †nn − 2∂ρ ξ n g †nρ    δSBV =  ξ ρ ∂ρ γab + 2∂(a ξ n βb) + 2∂(a ξ c γb)c †ab δg    δSBV =  ξ ρ ∂ρ βa + ∂n ξ n βa + ∂n ξ b γab + ∂a ξ n (−η 2 + βc β c ) + ∂a ξ b βb †na δg    δSBV =  ξ ρ ∂ρ (−η 2 + βc β c ) + 2∂n ξ n (−η 2 + βc β c ) + 2∂n ξ a βa †nn δg together with    δSBV =  ∂n (−η 2 + βc β c )g †nn + 2∂a (−η 2 + βc β c )g †na + 2∂(a βb) g †ab n δξ  +  2(−η 2 + βc β c )∂n g †nn + 2βa ∂n g †na + 2(−η 2 + βc β c )∂a g †na  +  2β(a ∂b) g †ab − Jab g †ab + ξ ρ ∂ρ χn + ∂ρ ξ ρ χn + ∂n ξ ρ χρ    δSBV = 2 ∂n βa g †nn + Jab g †nb + ∂b βa g †nb + ∂(b γc)a g †bc a δξ   1 †cd †nn †nb †nb †bc + 2 βa ∂n g βa ∂b g + γab ∂n g + γa(b ∂c) g − ∂a γcd g 2  − ∂a −η 2 + βc β c g †nn − 2∂a βc cg †cn + ∂ρ ξ ρ χa + ξ ρ ∂ρ χa + ∂a ξ ρ χρ   δSBV = ξ ρ ∂ρ ξ µ δχµ Now we would like to use these derivatives to write down the components of the bulk vector field Q, by imposing (46). We are using the antifields coordinates g †µν ,

16

A. S. CATTANEO AND M. SCHIAVINA

conjugate to gµν , and therefore we have to expand and reorder the terms as follows:   1 δS (Qg† )nn = − η −1  2 δη  1 −1 = − η Gη + ∂ρ ξ ρ g †nn + ξ ρ ∂ρ g †nn − 2∂ρ ξ n g †nρ 2    δS (Qg† )na = − β a (Qg† )nn 2 δβa   =Gβa + η −1 β a Gη + ∂c ξ c g †an + ξ ρ ∂ρ g †an − ∂ρ ξ a g †nρ − ∂c ξ n g †ac 2   δS ab (Qg† ) = + β a β a (Qg† )nn δγab    =Gγab − η −1 β a β b Gη + ∂ρ ξ ρ g †ab + ξ ρ ∂ρ g †ab − 2∂ρ ξ (a g †b)ρ 2 together with   δSBV (Qγ )ab = δg †ab  = ξ ρ ∂ρ γab + 2∂(a ξ n βb) + 2∂(a ξ c γb)c   δSBV (Qβ )a = δg †an  = ξ ρ ∂ρ βa + ∂n ξ n βa + ∂n ξ b γab + ∂a ξ n (−η 2 + βc β c ) + ∂a ξ b βb        −1 δSBV δSBV  −1 a b δSBV −1 a (Qη ) = − η + η β − η β β 2 δg †nn δg †na 2 δg †ab ρ n a n = (ξ ∂ρ η + ∂n ξ η − ηβ ∂a ξ ) and, with ρ = 1, 2, 3, n: (Qξ )ρ =



δSBV δχρ



 ;

(Qχ )ρ =

δSBV δξ ρ



The bulk Q vector field is extended to the normal jets when projected to the e pre-boundary vector field Q:  e µν = (∂n (Qgµν )) (QJ) = ∂n ξ ρ ∂ρ gµν + ξ ρ ∂ρ ∂n gµν + 2∂(µ ∂n ξ ρ gν)ρ + 2∂(µ ξ ρ ∂n gν)ρ ∂M ∂M of which we will only need e ab =  ∂n ξ ρ ∂ρ γab + ξ ρ ∂ρ Jab + 2∂(a ∂n ξ ρ gb)ρ + 2∂(a ξ c Jb)c + 2∂(a ξ n ∂n βb) (QJ)



so that the full pre-boundary vector field reads: e =(Q e η ) δ + (Q e β )a δ + (Q e γ )ab δ + (Q e g† )ab δ + 2(Q e g† )na δ Q †ab δη δβa δγab δg δg †na e g† )nn δ + (Q e ξ )n δ + (Q e ξ )a δ + (Q e χ )µ δ + ( Q e J )lm δ +(Q δg †nn δξ n δξ a δχµ δJlm e directly, we consider the folInstead of computing the explicit projection of Q lowing simplifying technique. We produce a degree one function via [20]: Se = ιQe ιEe ω e e is the Euler vector field on the space of pre-boundary fields, i.e.: where E Z δ δ δ e= E ξ ρ ρ − g †µν †µν − 2χρ δξ δg δχρ ∂M

BV-BFV GR: EINSTEIN HILBERT ACTION

17

Then the true boundary action S ∂ is such that Se = π ∗ S ∂ for degree reasons and the surjectivity of the surjection πM , which factors through FeADM . Moreover Q∂ is its Hamiltonian vector field. The boundary action is then found to be: ) Z (p        eab e ∂ 2 ∂ a †nn †na n en e e e J Jab − J + R − Λ − 2∂a ξ ge − 2e g ∂a ξ ξ S = γ e 4 ∂M ) Z (p  pγ   p e cd cd c †nb + γ e∂a Je − ∂c γ eγ e Jeda − (∂a γ e )Jecd + 2∂c ξe ge γ eba ξea 2 ∂M

(50) where by Je we denote the trace γ eab Jeab . Again using the definition of the Darboux coordinates, we can easily gather that the boundary action in that local chart will yield the expression (37), whereas the explicit components of Q∂ can be found using the equation ιQ∂ ω ∂ = δS ∂ . X This result is a clean first step in the direction of BV-BFV quantisation of General Relativity as proposed by CMR in [8]. It states the compatibility of bulk and boundary structures, in relation with the symmetries. Notice that the BV-BFV axioms 1.3 need not be satisfied by a generic gauge theory and the statement is therefore nontrivial. Arguable as it might be to consider gauge theories with this property to be somehow better quantisable, it provides nevertheless a clear mean of distinction between different variational problems describing the same equations of motion (see [15] for a comparison with the Palatini-Holst formulation of GR). The machinery is able to handle a more complex and sophisticated set of data, than the usual procedures of canonical analysis. When inducing (or not) a theory on the boundary it encodes a number of characteristic features, packing up relevant data in a very efficient way. As we will see in the following section, the piece of data that carries all the relevant information on the boundary is, non surprisingly, the boundary action. Finally, recall that in the 1 + 1 dimensional case it is known that the Einstein equations are trivial, and the symmetry distribution has to be amended to take conformal transformations into account. The critical dimension d = 1 is however marked out by the equations for the kernel of the pre-boundary 2-form ω e , both in the classical and the BV-extended case (cf. Theorem 2.1 and Proposition 2.1), confirming that the strategy has to be altered to analyse this specific example. 2.4. Constraint algebra and boundary gauge symmetry. As we already announced, from the boundary action (37) it is possible to read the constraint structure of canonical gravity. As a matter of fact, the degree zero (ghost number, gh) ∂ part of the derivatives δS δξ µ reads  √  δS ∂   ab 2 Π Π − Π + γ R∂ − 2Λ ≡ H = √ ab n δξ γ gh=0  δS ∂ = − ∂c γ cd Πda − (∂a γ cd )Πcd ≡ Ha a δξ

(51)

(52)

gh=0

which are the symplectic-reduced versions of the standard constraints (47) and (48). On the other hand, the residual gauge symmetries can be found by computing the relative components of the boundary cohomological vector field Q∂ , using the

18

A. S. CATTANEO AND M. SCHIAVINA

fact that ιQ∂ ω ∂ = δS ∂ : (Q∂ )ξn = (ξ c ∂c ξ n )

δ δξ n

(Q∂ )ξa = ξ n γ ab ∂b ξ n + ξ c ∂c ξ a

 δ δξ a

  (Q∂ )γ ab = ξ n Jeab + ξ c ∂c γ ab + 2∂(a ξ c γ b)c

δ δγ ab

It is interesting to notice that the symmetries above are a corrected version of the usual gauge symmetry for a d-dimensional metric on the boundary under the action of boundary diffeomorphisms ξ ∂ ∈ T [1]∂M . In fact they can be compactly rewritten as (Q∂ )γ =ξ n Je + Lξ∂ γ (Q∂ )ξa (Q∂ )ξn

1 =ξ n γ ab ∂b ξ n + [ξ ∂ , ξ ∂ ]a 2 =Lξ∂ ξ n

(53) (54) (55)

This means that they do not manifestly show a Lie algebra behaviour and the structure functions depend on γ −1 . Yet the boundary BFV action (37) is at most linear in the antighosts ϕ. This is in agreement with the observations in [14]. The BFV formalism provides for a cohomological resolution of symmetry-invariant coisotropic submanifolds [21, 22, 3, 19, 6], and in this case of the constraint submanifold of canonical gravity, modulo residual gauge symmetry. The (cohomological) description of the the canonical, constrained phase space for General Relativity is then obtained from a simple variational problem in the bulk. This encompasses a number of classical results in the field while clarifying related issues at the same time. Moreover, we stress that on top of obtaining the expected BFV resolution of the canonical structure on the boundary, we are able to establish a connection with the boundary data through the explicit projection π, BV and the fundamental equation ιQ Ω = δSADM α∂ . This is the starting point for the BV-BFV programme to quantisation of gauge theories on manifolds with boundary. References [1] A.S. Cattaneo, P. Mnëv, N. Reshetikhin, Classical BV theories on manifolds with boundary, Comm. Math. Phys. 332 (2): 535-603 (2014). [2] I. A. Batalin and G. A. Vilkovisky. Gauge algebra and quantization, Physics Letters B 102.1 (1981): 27-31. [3] I. A. Batalin, E. S. Fradkin, A generalized canonical formalism and quantization of reducible gauge theories, Phys. Lett. B 122 2, 157-164 (1983). [4] I. A. Batalin, G. A. Vilkovisky, Relativistic S-matrix of dynamical systems with boson and fermion costraints, Phys. Lett. B 69 3, 309-312 (1977). [5] C. Becchi, A. Rouet and R. Stora, Phys. Lett. B52 (1974) 344. C. Becchi, A. Rouet and R. Stora, Commun. Math. Phys. 42 (1975) 127. C. Becchi, A. Rouet and R. Stora, Ann. Phys. 98, 2 (1976). I. V. Tyutin, Lebedev Physics Institute preprint 39 (1975), arXiv:0812.0580. [6] J. Stasheff, Homological reduction of constrained Poisson algebras, J. Diff. Geom. 45, 221-240 (1997). [7] A.S. Cattaneo, P. Mnëv, N. Reshetikhin, Semiclassical quantization of Lagrangian field theories, Mathematical Aspects of Quantum Field Theories, in Mathematical Physics Studies 2015, pp 275-324. [8] A.S. Cattaneo, P. Mnëv, N. Reshetikhin, Perturbative quantum gauge theories on manifolds with boundary, arXiv:1507.01221. [9] M. Atiyah, Topological quantum field theories, Inst. Hautes Etudes Sci. Publ. Math. 68. 175-186 (1988).

BV-BFV GR: EINSTEIN HILBERT ACTION

19

[10] G. Segal, The definition of conformal field theory, in: Differential geometrical methods in theoretical physics, Springer Netherlands, 165-171 (1988). [11] P.A.M. Dirac, Generalized Hamiltonian dynamics, Canad. J. Math. 2, 129-148 (1950). [12] B.S. DeWitt, Quantum Theory of Gravity. I. The Canonical Theory, Phys. Rev. 160, 1113, (1967). [13] R. Arnowitt, S. Deser, C. Misner, Dynamical Structure and Definition of Energy in General Relativity. Physical Review 116 (5), 1322-1330 (1959). [14] C. Blohmann, M.C. Barbosa Fernandes and A. Weinstein Groupoid symmetry and constraints in general relativity, Commun. Contemp. Math. 15, 1250061 (2013). [15] A.S. Cattaneo and M. Schiavina, BV-BFV approach to General Relativity. Part 2: PalatiniHolst action, work in progress. [16] A.S. Cattaneo and M. Schiavina. On time, work in progress. [17] P. Mnev, Discrete BF theory, arXiv:0809.1160 (2008). [18] G. Felder and D. Kazhdan. The classical master equation, arXiv:1212.1631 (2012). [19] J.Stasheff, Deformation Theory and the Batalin-Vilkovisky Master Equation, Deformation theory and symplectic geometry, proceedings, Meeting, Ascona, Switzerland, June 16-22, 1996, arXiv:q-alg/9702012. [20] D. Roytenberg, AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories, Lett. Math. Phys. 79: 143-159 (2007). [21] F. Schaetz, BFV-complex and higher homotopy structures, Comm. Math. Phys. 286 (2), 399-443 (2009). [22] F. Schaetz, Invariance of the BFV complex, Pac. J. Math. 248 (2), (2010). institut für Mathematik, Winterthurerstrasse 190, 8057 Zürich, Switzerland E-mail address: [email protected] institut für Mathematik, Winterthurerstrasse 190, 8057 Zürich, Switzerland E-mail address: [email protected]