BRST BMS4 Symmetry and its Cocycles from Horizontality Conditions

The BRST structure of the extended Bondi-Metzner-Sachs symmetry group of asymptotically flat manifolds is investigated using the recently introduced framework of the Beltrami field parametrization of four-dimensional metrics. The latter identifies geometrically the two physical degrees of freedom of the graviton as fundamental fields. The graded BRST BMS4 nilpotent differential operator relies on four horizontality conditions giving a Lagrangian reformulation of the asymptotic BMS4 symmetry. A series of cocycles is found which indicate the possibility of anomalies for three-dimensional Lagrangian theories to be built in the null boundaries of asymptotically flat spaces from the principle of BRST BMS4 invariance.


Introduction
This paper analyses the BRST structure of the infinite-dimensional asymptotic symmetry group of asymptotically flat four-dimensional Lorentzian manifolds M 4 .This group was discovered in the early 60's by Bondi, Metzner, van der Burg [1] and Sachs [2] and it is currently denoted as the BMS4 group.It contains the Poincaré group as one of its subgroups as well as an infinite-dimensional extension of the translations, known as the supertranslations.It can be further enlarged to also contain the so-called superrotations that are an infinite-dimensional generalization of the Lorentz transformations and play a key role in the context of flat space holography [3,4].One then gets the extended BMS4 group.Bigger asymptotic symmetry groups were later on discovered by relaxing the falloff conditions of asymptotically flat spaces and/or softening part of the Bondi gauge conditions as discussed in [5,6,7].
The seminal work of Strominger [8] has inspired deeper studies of the BMS symmetry.In particular, [9] exhibits links between the Ward identities for the supertranslation symmetry, the Weinberg soft graviton theorem and the gravitational memory effects [10] for gravity.Analogous links also exist for QED.Such relations between three apparently uncorrelated area of physics can be pictured as "infrared triangles" (see [11] for a review).They have been generalized by including the superrotation invariance in [12,13], that gives relations between the subleading soft graviton theorems [14] and a description of spin memory effects [15].[16] reviews these recent developments that also involve studying the charge algebra of the presently known versions of the BMS symmetry within the covariant phase space formalism [17,18].It is noteworthy that the algebra of these Hamiltonian charges may exhibit a central extension related to the existence of a non trivial 2 cocycle for a BRST BMS Hamiltonian operator [19].
The goal of this paper is actually to properly construct the Lagrangian BRST operation for the BMS4 symmetry and determine the possible cocycles that may control the model dependent anomalies of three-dimensional local quantum field theories constructed from the principle of the BRST BMS4 invariance in the null boundaries I ± of M 4 .This is done within the recently introduced leaf of leaf foliation scheme [20].The latter generically refines the ADM foliation [21] and provides a covariant and local decomposition of the ADM leaves Σ d−1 = Σ 2 × Σ d−3 of Lorentzian manifolds M d by randomly varying surfaces Σ 2 .This description is analytically expressed by a so-called generalized Beltrami parametrization of d-beins in terms of d(d + 1)/2 local fields, referred to as d-dimensional Beltrami fields, which suggestively generalize in higher dimensions the wellestablished 3 Beltrami fields µ z z , µ z z and Φ that covariantly parametrize the bidimensional Beltrami metric ds 2 = exp Φ||dz + µ z z dz|| 2 .The bidimensional Beltrami parametrization has proven to be a very useful tool to study many aspects of 2d gravity and supergravity since the 80's [22,23,24,25].It is certainly rewarding that its generalization for d > 2 (d = 4 in this paper) provides new suggestive perspectives on the various metric components while respecting the covariance principles and taking care of the propagation of physical gravitons.Constructing the generalized d > 2 covariant Beltrami parametrization in [20] was actually motivated by the search for a covariant decomposition of the d-dimensional metric components into distinct sectors, each of them having its physically relevant interpretation.It was found that the physical d(d − 3)/2 degrees of freedom of massless ddimensional gravitons can be identified within the leaf of leaf metric decomposition as a well-defined subset of the d(d + 1)/2 Beltrami fields, geometrically identified as the fundamental gravitational fields.
If one focuses on the null boundaries of four-dimensional asymptotically flat space, the surfaces Σ 2 can be identified with the celestial spheres foliated along their null directions.The excitations of their Beltrami differential µ z z and µ z z stand for the field description of the two physical helicity states of a graviton, as will be shown in great details in a different paper.The definition of the BRST symmetry associated to the four-dimensional BMS4 group, generically denoted as the BRST BMS4 symmetry in this paper, will be built through geometrical horizontality conditions within the context of the d = 4 Beltrami parametrization.
It will be shown that BRST BMS4 descent equations exist and define cocycles, that indicate the existence of three-dimensional anomalies for theories defined at the tree level from the principle of BMS4-BRST invariance in I ± .Getting what can be called the fundamental space representation of the BMS4 algebra is actually a quite transparent task from the point of view of the BRST BMS4 symmetry, as it can be geometrically defined.Classifying all its higher space representations is a very non trivial question.
It must be noted that the BRST Beltrami construction presented in this work is quite appropriate to construct supersymmetric extensions of the BMS group.Such structures have been called super-BMS group [26,27].This could be done by extending the genuine pure gravity horizontality conditions used in this work to those of topological gravity before twisting it to get N = 1 supergravity.The way it goes is to be presented in a different publication.The Beltrami construction also clarifies other aspects of the graviton emissions and absorptions near the null boundaries of spacetime, as well as their relation with the vacuum changes of the celestial sphere, to be also published in another publication.
The rest of the paper is organized as follows.Section [2] recalls the basic ingredients of the four-dimensional Beltrami parametrization and adapts it to the specific falloff conditions of asymptotically flat spacetimes.
Section [3] explicitly constructs the nilpotent BRST BMS4 operator as stemming from fourdimensional horizontality conditions.The way this operator acts on the data at null infinity and on the basic ghost fields of the theory is exposed.
Section [4] exposes a non trivial cocycle ∆4 , solution of the Wess and Zumino consistency conditions (s + d) ∆4 = 0 where s is the local nilpotent differential operator that represents the BRST BMS4 symmetry.The explicit form of ∆ 1  3 is discussed.Its existence indicates possible model dependent breakings of the Ward identity of the BRST BMS4 symmetry, which can possibly invalidate some of the local 3-dimensional quantum field theories one may consider in I + .Finally, appendices [A] and [B] respectively compute the four-dimensional Spin Connection and the BRST transformations of the ten Beltrami fields, whose detailed expressions are necessary ingredients for the work presented here.
2 Four-dimensional Beltrami Parametrization of Asymptotically Flat Manifolds 2.1 Beltrami Parametrization [20] defines the covariant Beltrami parametrization of a generic vierbein as follows where ϕ = ϕ ≡ Φ 2 .(2.1) is the generalization of the Beltrami zweibein e z ≡ exp Φ 2 (dz + µ z z dz), e z ≡ exp Φ 2 (dz + µ z z dz) as it was defined in [22] in the context of string theory.It provides a covariant and local decomposition of the ADM leaves Σ 3 = Σ 2 × Σ 1 of Lorentzian manifolds M 4 by randomly varying surfaces Σ 2 .Here (z, z) are complex coordinates for Σ 2 , x 0 ≡ t is for the real spatial transverse direction for this set of coordinates and τ is for the Lorentz time.For asymptotically flat manifolds, Σ 2 is a Riemann sphere at spatial infinity and the Minkowski metric writes with γ zz = 4 (1+zz) 2 , so that z and z are complex coordinates on the unit sphere.In polar coordinates γ zz dzdz = dθ 2 + sin(θ) 2 dϕ 2 .The Beltrami line element is recovered by using (2.1) and computing g µν = e a µ η ab e b ν , giving where a ≡ µ 0 τ , with a slight change of notations as compared to [20].The metric (2.3) or, equivalently, the vierbein (2.1), defines the ten four-dimensional "Beltrami fields" M, N, Φ, µ 0 τ , µ z z , µ z z , µ z 0 , µ z 0 , µ z τ , µ z τ in coordinates (τ, t, z, z), analogously as the bidimensional formula ds 2 = exp Φ||dz + µ z z dz|| 2 defines the standard Beltrami differential µ z z and the conformal factor Φ for a Riemann surface.The seven fields µ a b are Weyl invariant while M, N, Φ transform non trivially under Weyl transformations.
The definition (2.1) is supported by a Diff 4 ⊂ Diff 4 ×Lorentz covariant gauge fixing of the 16 components of any given generic vierbein into the ten Beltrami fields that parametrize both the vierbein and the d = 4 metric in (2.1) and (2.3).[20] indicates that these 6 gauge functions split into the 5 "Beltrami conditions" e 0 z = e 0 z = e τ z = e τ z = 0, a ≡ µ 0 τ = −µ τ 0 and a 6th one, imposing that e z and e z have the same conformal factor exp Φ.After such a Lorentz gauge fixing, the Lorentz ghosts get fixed as functions of the reparametrization ghosts, so that the reparametrization BRST invariance of the "Beltrami conditions" is warranted.Appendices[A] and [B] compute their expressions as well as that of the Spin connection ω(e) that solves the condition T = de+ω(e)∧e = 0 and its consistency condition DT a = R ab ∧ e b = 0, in function of the Beltrami fields.
The use of light cone coordinates τ ± = τ ± t is a must in the framework of the BMS symmetry and of the gravitational waves.The light cone Beltrami metric equivalent to (2.3) is and the corresponding Beltrami light cone vierbein is (2.5) The consistency between the two sets of coordinates in (2.3) and (2.4) implies that the fields (Φ, µ z z , µ z z ) remain the same when one passes from one system to the other and that one has where the fields in the r.h.s. of these equalities are functions of (z, z, t, τ ).

Bondi Gauge and Falloff of the Beltrami Fields
For the rest of this article, attention will be focused on future null infinity I + , although all of the results are easily adapted to I − .The Bondi metric near I + in retarded Finkelstein-Eddington coordinates (u ≡ τ − , r ≡ t, z, z) is often denoted as [2] Here A = z, z or A = θ, φ refer to the polar coordinates on the possibly distorted celestial sphere.The so-called Bondi gauge functions are and the falloff conditions of asymptotically flat spacetimes that solve Einstein's equations are , BMS [1,2] have shown that (2.7) and (2.9) provide a most practical and precise description of gravitational plane waves near the null boundaries of M 4 .One commonly calls m B (u, z, z) the Bondi mass aspect and C AB the traceless shear tensor that describes gravitational waves (the traceless property comes from the determinant gauge condition).The background metric γ AB is conformally related to the metric of the unit 2-sphere as γ AB dx A dx B = γ zz exp Φdzdz.
It turns out that the Bondi gauge is simply recovered by imposing the three gauge conditions in the Beltrami light cone metric (2.4).Indeed, by going to the retarded Finkelstein-Eddington coordinates by imposing τ + = u + 2r, the metric rewrites (2.11) The Bondi metric (2.7) is thus recovered in the Beltrami parametrization, according to (2.12) (2.10) can thus be understood as a gauge fixing to zero of the shift vector of a mini foliation, near I + and along the dτ + direction, which is the direction of possibly absorbed gravitons in I + .One immediately infers that this gauge fixing of the metric is to be incomplete, analogous to a Yang-Mills gauge fixing in the temporal gauge, leading one to zero modes in the reparametrization Faddeev-Popov ghosts.Their study within the d = 4 Beltrami parametrization and its BRST symmetry is actually the subject of this work * .
Each Beltrami field X can be expanded near I + as a Laurent series in function of r with the following notation X(u, r, z, z) ≡ p r −p X −p (u, z, z). (2.13) The first line of (2.12) implies that the term of order 1 in the 1/r expansion of µ + − can be identified with the Bondi mass aspect, according to µ z z , µ z z and Φ build a standard Beltrami parametrization of the 3 degrees of freedom of the bidimensional matrix g AB , such that so that one has (2.16) If one uses the (z, z, t, τ ) system of coordinates, (2.6) implies that the gauge choice (2.10) is where τ − ≡ u.The determinant gauge condition is . (2.18) At first non trivial order O(r −1 ), (2.18) implies This constraint is to play an important role when combined with the Beltrami field falloff conditions near I + .In fact, the latter are obtained by plugging the definition of the Beltrami metric (2.11) within the falloff conditions (2.9).The asymptotic Beltrami fields are to satisfy the following 1/r expansions for large values of r Φ −1 is set to zero in (2.20) because of (2.19) and µ z z 0 = µ z z 0 = 0, due to the falloffs of µ z z and µ z z .This is a major simplification when building the BRST operator associated to the BMS symmetry.

Extended BMS4 Algebra from BRST Horizontality Conditions
The extended BMS4 symmetry is defined as the residual Diff 4 symmetry that leaves invariant the Bondi gauge (2.10) and the falloffs (2.20).The existence of the latter has been justified by the analogy between (2.10) and the Yang-Mills temporal gauge.The associated BRST symmetry operation for this restricted Diff 4 symmetry needs introducing its (Faddeev-Popov) ghosts as anticommuting vector Diff 4 reparametrization ghosts ξ µ (x), with an x dependence restricted by the four consistency conditions for the Bondi gauge at all order in the 1/r expansion and, for the falloff conditions, by (3.1) can be seen as the equations of motion of the antighosts stemming from the BRST exact terms that ensure the 4 gauge functions (2.10).They are the generalization of the degeneracy of the Yang-Mills ghost equation of motion when one uses the temporal gauge A τ = 0, which provides sA τ = ∂ τ c = 0. Getting degenerate Faddeev-Popov ghosts is often the consequence of using so-called "physical gauges".
By relaxing the conditions on sµ z z and sµ z z , that is allowing for non zero µ z z 0 and µ z z 0 in the background metric γ AB would lead to what is known as the generalized BMS4 algebra.The reasoning that follows could also be applied in this more general case, but brings no real extra information due to the zero genus of the celestial sphere.
Appendix[B] computes the BRST transformations sX Beltrami of the ten Beltrami fields X Beltrami from the ghost dependent geometrical torsion free equation that encodes all the Diff 4 BRST symmetry [29] T where ξ µ (x) is the anticommuting reparametrization ghost, ω ab (x) and Ω ab (x) the Lorentz Spin connection and ghost respectively and ẽa ≡ exp i ξ e a .This permits one to express the consistency conditions (3.1).The found nilpotent BRST operator s acting on the Beltrami fields is in fact equivalent to that acting on a generic metric g µν with The four consistency conditions (3.1) of the Bondi gauge (2.8) are therefore Here, exp Φ ′ ≡ r 2 γ zz exp Φ and the indices i and A run respectively over (u, z, z) and (z, z).(D z , D z ) stand for Using (2.18), one can solve the equations (3.5) that are valid to all order in the 1/r expansion.It provides the following metric dependent restrictions for ξ µ (u, r, z, z): At first non trivial order in 1/r one gets ∇ A is the covariant derivative with respect to the unit 2-sphere, giving for example The subset of the Diff 4 reparametrization symmetry with infinitesimal parameters represented as in (3.7) defines the leftover gauge invariance of the gravitational action expressed in the Bondi gauge.This proves that it doesn't provide a full gauge fixing, as announced.
Let us now take into account the falloff restrictions (3.2), which is to reduce further the coordinate dependence of the ghost zero modes.At leading order in the Bondi gauge, one can rewrite (B.31), (B.32) and (B.33) of Appendix[B] as One must observe at this point that the s transformations of the Beltrami fields are computed in the coordinate system (τ, t, z, z) in Appendix [B].Therefore, one must carefully compute the coordinate derivations on the fields in (2.17), consistently with the change of coordinates (τ, t, z, z) → (u, r, z, z), that is One must also redefine the vector ghosts that are defined in Appendix[B] as The restrictions on the ghosts ξ µ given by (3.8) and the falloffs (2.20) of the Beltrami fields can then be safely combined to enforce the falloff consistency conditions (3.2) on (3.9)-(3.11).One gets where α(z, z) is an arbitrary given function of (z, z) obtained by a trivial quadrature over the u coordinate.Analogously, one gets ξ z 0 = ξ z 0 (z).At this stage, one can simply check that all remaining constraints associated to (3.2) are satisfied due to (3.8) and (3.14).It follows that the residual symmetry of Diff 4 in the Bondi gauge with the appropriate falloffs (2.9) is carried by the restricted four ghosts ξ µ BMS , that are actually well defined functionals of a "fundamental geometrical ghost representation" made of the following anticommuting fields α(z, z) , One can in fact use the following notation The BMS beautiful result is of course recovered by expanding in spherical harmonics the arbitrary function α(z, z), namely the non trivial asymptotic d = 4 reparametrization invariance in the Bondi gauge includes not only the obvious spatial and time translations through the lowest harmonics (l = 0, 1) but the so-called supertranslations also arise as the higher order harmonics l, m for l > 1, m = −l, ..., l. ξ z 0 and ξ z 0 can be identified as solutions of a two-dimensional conformal Killing equation on the two-dimensional celestial sphere.As noted elsewhere, it is well known that the set of globally well defined solutions to this equation on S 2 is six-dimensional, and the resulting algebra is equivalent to that of the d = 4 Lorentz group.But if one allows local solutions with singularities, the set of solutions is enhanced to form the infinite-dimensional Virasoro algebra which generates the superrotations.In this case, (3.15) generate the so-called extended BMS4 algebra.
The defining horizontality conditions (3.3) actually contains a built in representation of this algebra through the action of the BRST operator s on the primary ghosts (3.15).In fact, rewriting (B.34) and (B.35) at leading order in the Bondi gauge leads to That is Using the same technique for sc z and decomposing ξ u as in (3.17), one gets the nilpotent BRST operator of the extended BMS4 algebra: Note that this algebra is realized at the level of the ξ µ BMS given by (3.17) when equipped with the modified BRST operator s ≡ s − δ g ξ where δ g ξ was introduced in [4].It measures the variation of the ξ BMS 's under a diffeomorphism due to their explicit dependence in the metric and acts as δ g ξ g µν = L ξ g µν on the metric.An important point is understanding the action of a BMS transformation on the relevant data at null infinity.In the Bondi language, it amounts to compute the transformation laws of the shear tensor and its derivatives under the just built BRST-BMS operator.In terms of Beltrami fields, the shear tensor writes This equation is one of the many indications that the fields µ z z −1 and µ z z −1 stand for field representations of the two helicity states of the graviton.More on that will appear in a different publication.
The s transformation of the shear is given by the term of order O(r −1 ) in (3.9), that is which eventually gives when the freedom on the determinant gauge condition (ω ′ in (3.2)) is used to fix the conformal factor exp Φ such that γ zz exp Φ = 1 + O(r −2 ).The transformation sµ z z −1 is of course analogous to this one.
The above computed values of the transformation of the Beltrami differential sµ z z −1 and sµ z z −1 in (3.23) and of the primary ghosts ξ z 0 , ξ z 0 and α in (3.20) coincide with the transformations of the BMS4 algebroid as it is computed by Barnich in the Hamiltonian formalism [19].
What differs in our derivation is that it directly and unambiguously computes the nilpotent BRST symmetry for the BMS4 algebra from the geometrical four-dimensional horizontality conditions (3.3), which express the nilpotent BRST symmetry for the full Diff 4 , suitably constrained by the four Bondi gauge conditions and the falloff conditions of the ten Beltrami fields.
Getting this straight four-dimensional derivation, allowed by the use of the Beltrami parametrization, is an important result of this work.We actually expect that the Beltrami formalism is going to be very helpful for handling the non trivial situation of computing the BMS structure when local supersymmetry is involved, through the horizontality conditions of topological gravity with further twists.This will appear in a separate publication.
Having clarified the BRST structure of the BMS4 symmetry allows us to pass to the next section devoted to the search of non trivial solutions of the Wess and Zumino equations for the BMS4 symmetry, that is for the search of non trivial consistent cocycles for the s symmetry, postulated as a fundamental symmetry for defining relevant three-dimensional quantum field theories on the boundaries I ± of M 4 .

Non Trivial Cocycles of the Nilpotent BRST BMSAlgebra
The BRST BMS4 symmetry is a truncation of the full Diff 4 BRST symmetry with restricted reparametrization ghosts.One might call the 3 ghosts ξ z (z), ξ z (z), α(z, z) its "fundamental" ghost field representation, defined by (3.20).
The Diff 4 BRST symmetry has no non trivial cocycles as can be proven by showing their equivalence with hypothetical non trivial ones for the local Lorentz symmetry.Indeed, the latter cannot exist for SO(1, 3) (more generally the Lorentz anomalies can only exist for d = 2 modulo-4).However, the cocycle equations for the BRST BMS4 symmetry operator s are less demanding than those for the full SO(1, 3)×Diff 4 symmetry and nothing but a computation can confirm if they have or not non trivial solutions.It is a mathematically legitimate question to look for such solutions.
Moreover, a better understanding of the gravitational boundary effects suggests building threedimensional local quantum field theory in I + governed by the nilpotent BRST BMS4 symmetry as constructed in the last section, a bit analogously as one builds string theory as a bidimensional theory with the remaining symmetry of the Diff 2 of the worldsheet in the conformal gauge, namely the modular invariance.The physical motivation for computing the cohomology of the BRST operation s at various ghosts numbers is for checking if quantum anomalies may occur.Therefore, one must check if a local 3-form cocycle with ghost number 1 exists, generically denoted as ∆ 1  3 (µ z z −1 , µ z z −1 , ξ z (z), ξ z (z), α(z, z)).∆ 1  3 must be such that a 2-form ∆ 2 2 cocycle with ghost number 2 exists and is solution of and so on until a possibly s invariant ∆ 0 4 cocycle that is not s exact.∆ 1  3 being a non trivial cocycle means that both ∆ 1 3 and ∆ 2 2 are defined modulo s and d exact terms.If such a cocycle exists, I + ∆ 1  3 may appear in the right hand side of the Lagrangian BRST BMS4 Ward identity times a non zero coefficient a whose value depends on the chosen QFT in I + , without being possibly absorbed in the effective 3-dimensional action by appropriate changes in its renormalization procedure.This would thereby invalidate the chosen 3-dimensional local quantum field theory that one would consider in I + based on the principle of s invariance.

A Bidimensional Motivation
To gain intuition on the way to compute non trivial cocycles for the BRST BMS4 symmetry, one may firstly consider the nilpotent ghost transformations (3.20) of its basic ghost ξ z 0 (z), ξ z 0 (z) and α(z, z).As such, these fields can be considered as genuine bidimensional objects, with no reference whatsoever to the third and fourth coordinates r and u.
One may indeed consider a quite simple bidimensional structure, based on the holomorphic and antiholomorphic ghosts ξ z 0 and ξ z 0 .Denoting m z z (z, z) ≡ µ z z 0 and m z z (z, z) ≡ µ z z 0 , one can define the generalized anticommuting 1-forms on the celestial sphere M z 0 ≡ dz + m z z (z, z)dz + ξ z 0 (z) and M z 0 ≡ dz + m z z (z, z)dz + ξ z 0 (z).Part of the intuition for doing so is that m z z (z, z) can be interpreted as the Beltrami differential component µ z z 0 (z, z) on the celestial sphere, which has been consistently chosen equal to zero in our construction of the extended BMS4 symmetry.
The action of s acting on all field components of M A is defined as the following horizontality conditions By expansion in ghost number, one recovers keeping in mind that these fields are taken equal to zero elsewhere in the paper.These transformation laws are indeed the same than the one for the zero order component of the metric, µ z z 0 and µ z z 0 , when c A ≡ ξ A 0 .Therefore, these equations remain consistent with sµ z z 0 = sµ z z 0 = 0 when µ z z 0 = µ z z 0 = 0.So (4.2) consistently defines the action of the BRST BMS4 operation s on ξ z 0 and ξ z 0 as well as on µ z z 0 and µ z z 0 .
The nilpotency property s 2 = 0 trivially holds true as a graded nilpotent differential operation due to the Jacobi identity of the Poisson bracket and to the chosen grading properties of all fields.
One may tentatively introduce an horizontality condition for the s transformation of α, by introducing the following 1-form Ũ ≡ du + U z (z, z)dz where and λ is any given real number.Both components of the 1-form U , U z (z, z) ≡ µ u z 0 and U z (z, z) ≡ µ u z 0 , are defined on the celestial sphere with no u dependence.U can be understood as a BRST antecedent of α, according to U → U + α.

Quite remarkably, the part of ghost number 2 of the horizontality condition
3) provides the 4-dimensional formula (3.20) for λ = 1 2 .The part with ghost number 1 of G = G defines the s transformation of the BRST antecedent U of α, as follows One can check that s 2 = 0 on α, U z and U z for all values of λ.In fact, this property is directly implied by the Bianchi identity since by using G = G, one has identically denoting d = s + d.The terms of ghost number larger or equal to 2 of this equation then implies s 2 Ũ = 0.
The choice λ = 1 2 fixes the conformal weight of α on the celestial sphere.It must be understood that this value is determined by using all ingredients of the Diff 4 symmetry near I + , while the λ dependent formula (4.3) are genuinely bidimensional and covariantly well defined on the celestial sphere with du considered as a 0-form, see [30].
The strength of (4.2) and (4.3) is its simplicity for defining the s operation acting on α and ξ A 0 , and identifying their BRST antecedent µ u z 0 , µ u z 0 , µ z z 0 and µ z z 0 .However, we have not been able to reexpress the action of the BRST BMS4 symmetry on the graviton fields µ z z −1 and µ z z −1 as stemming from purely bidimensional horizontality equations analogous to (4.2).The difficulty originates from the terms ( in the full four-dimensional formula (3.23).Finding this term would require to modify (4.2) as where the generalized 1-forms are now defined as The term with ghost number one, proportional to dz and of order O(r −1 ) would then imply the correct s transformation for µ z z −1 , but the terms with ghost number 2 would break the transformation law sξ A 0 of the primary ghosts.The above results indicate the deep four-dimensional origin of the BMS symmetry of I + , despite its apparently simple two and three-dimensional structure.As a matter of fact, the celestial holography program [31] could encounter difficulties due to the necessity of making it consistent with the complete Diff 4 invariance of the bulk.
However, the bidimensional approach (4.2) suggests the following method for computing non trivial BRST BMS4 cocycles involving the graviton field µ z z −1 and µ z z −1 .One first observes that, in terms of the not so relevant m z z = µ z z 0 and m z z = µ z z 0 , one can define the following unified invariant cocycle for the d + s operation The ghost expansion of (4.9) provides four consistency equations s∆ g 4−g(0) + d∆ g+1 3−g(0) = 0 with ghost numbers 1, 2, 3 and 4 within I + , such that the ∆ g 4−g(0) 's are defined modulo d and s exact terms.For example, ∆ 1 3(0) decomposes as follows The three-dimensional s consistency equation s I + ∆ 1 3(0) = 0 indicates that one may possibly get an anomaly for d = 3 quantum field theories built within I + from the principle of the s invariance, having set µ z z 0 (z, z) = 0 in the derivation of the BMS4 symmetry.On the other hand, the u independence of µ z z 0 and µ z z 0 makes the finality of the above exercise a bit formal, but the latter suggests how to find non trivial cocycles solution for the obviously more interesting case of finding anomalies for d = 3 quantum field theories involving the asymptotic graviton fields µ z z −1 and µ z z −1 .

Computing the BMS4 Cocycles in I +
As a matter of fact, what concretely matters is the possibility of an anomaly occurring when computing the various entities relative to the production or absorption of gravitons in the celestial sphere, whose fields are represented by µ z z −1 and µ z z −1 , while respecting all the consequences of the BMS4 symmetry, in particular for all conservation laws, including of course the celebrated Bondi mass loss formula (to be reformulated in a separate paper using the Beltrami formulation).
One is thus looking for a series of s-cocycles ∆ g 4−g , g = 1, 2, 3, 4, possibly generated by (4.1), which satisfy and are a priori dependent (i) on the three ghosts ξ A 0 and/or α, to possibly carry the ghost number g = 0 of ∆ g 4−g , and (ii) on µ z z −1 and µ z z −1 .One assumes that ∆ 1 3 is made of exterior products of dz, dz and du in order that it is a well defined form on I + .Solving (4.12) is of course non trivial for µ z z −1 and µ z z −1 dependent cocycles, since the BRST BMS4 transformation law (3.23) of the former fields seems not to be expressible as part of a genuine horizontality condition implying simple descent equations.
However, if one manages to compute ∆ 1  3 ∝ dudzdz with a linear ghost dependence in ξ A 0 and/or α to carry ghost number one, and satisfying the algebraic Poincaré lemma implies the existence of a 2-form ∆ 2 2 such that and then using the now obvious equation ds∆ 2 2 = 0 and so on, one gets successively the existence of non trivial cocycles satisfying Therefore, everything boils down to guessing a possible form for ∆ 1 3 .Going back to the above µ z z 0 , one can observe that, not only its derivation was suggestive, but the way the conformal indices get contracted to get a 3-form is quite unique.It is indeed expected that a possible anomaly ∆ 1  3 should be linear in the graviton fields as ∆ 1  3(0) .This makes it quite natural to check if the following equation is valid: To do so, one must uses (3.20), (3.23), and, quite amazingly, one finds that sd(∆ 1 3 ) = 0 and thus (4.15) follows.
Getting the explicit expressions of the cocycles ∆ g 4−g for g > 1 is rather hard work in the absence of horizontality equations.As a matter of fact, (4.14) and (4.15) are satisfied for with the simplified notations The last cocycle satisfies s∆ 4 0 = 0.The complicated expressions of the lower descendant cocycles ∆ 2 2 , ∆ 3 1 and ∆ 4 0 contrast with the simplicity of their generating ghost number one up ascendant ∆ 1  3 = du∧dz∧dz (µ∂ 3 ξ + µ∂ 3 µ), as (correctly) postulated in (4.16).As a matter of fact, the simplicity of the quadratic field dependence of ∆ 1 3 makes it rather obvious to check that it is neither d or s exact.Hence, one has the same property for the other cocycles ∆ g 4−g .So, the existence of ∆ 1  3 is a clear signal that one may have an anomaly in the context of three dimensional Lagrangian quantum field theories in I + , relying on the principles of the BRST BMS4 invariance.
In practice, a I + ∆ 1 3 (ξ, ξ, µ, µ) with a = 0 is an obstruction for imposing the Ward identity of the BRST BMS4 symmetry of the chosen quantum field theories in I + .The study of this possibly broken Ward identity permits one to identify which correlation functions must be investigated to compute the value of the model dependent anomaly coefficient a. Physically, having a = 0 may complicate the construction of theories describing graviton absorptions and creations in the null boundaries I ± of asymptotically flat manifolds.
Alternatively, if one uses a Hamiltonian formalism, the coefficient a is to be interpreted as a central charge.The breaking of the Hamiltonian BRST BMS4 Ward identity implies that of the nilpotency of its BRST charge Q, as shown e.g. by Kato and Ogawa [32] in the context of the Polyakov covariant bosonic string theory.One actually expects Q 2 = a S 2 ∆ 2 2 in operational form.It is noteworthy that the ∆ g 4−g 's for g ≥ 2 correspond to the BMS4 algebroid cocycles that Barnich computed in [19].However, the Hamiltonian approach leaves undetermined the generating Lagrangian top cocycle ∆ 1  3 .

Appendices A Computation of the d = 4 Beltrami Spin Connection
This appendix exposes the full form of the spin connection ω(e) for d = 4 Beltrami gravity as the solution of 24 linear equations.They are the result of the vanishing torsion conditions T τ = T 0 = T z = T z = 0 where T a = de a + ω a b ∧e b and will be developed here.One works with the Beltrami parametrized vierbein (2.1), namely where e z = exp ϕ E z , e z = exp ϕ E z and a ≡ µ 0 τ , a ≡ µ τ 0 .The vanishing torsion equations are first solved in the general case ϕ = ϕ and a = a, but this Lorentz gauge fixing will be enforced at the end of the computations.The following equations are slightly different from the one derived in [20] as one uses the Minkowski metric (2.2) There are 12 equations that come from the 2-form decomposition of T τ = T 0 = 0.This decomposition will be done on the basis (E z , E z , dt, dτ ).In this basis, the exterior derivative takes the form and the spin connection decomposes as The expressions of the curly derivatives D are recovered by using (A.1), which gives The 12 equations coming from T τ = T 0 = 0 are then pretty easy to derive, this goes as which gives the first 6 equations.The others come from the same kind of decomposition for T 0 , that is To find the 12 remaining equations, one must do the same 2-forms expansion for T z and T z .This is slightly more involved than doing it for T τ and T 0 because some relations are needed to relate the components of forms expressed either on the basis of 1-forms (dz, dz, dt, dτ ) or on the basis (E z , E z , dt, dτ ).This is done by rewriting all products of forms involving dz and/or dz in terms of the elements of the basis (E z , E z , dt, dτ ).By using (A.1), it yields to Then, T z and T z read The condition T z = 0 is then from which 6 other linear equations are deduced.
Analogously, the condition T z = 0 writes from which one deduces the last 6 equations that are necessary to compute ω(e).
The full system of 24 linear independent equations, made of (A.6), (A.7), (A.11) and (A.12), can be solved numerically for a = −a and ϕ = ϕ = Φ 2 .One then obtains the expression of the 24 components of the spin connection ω(e) when the vierbein is expressed in its Beltrami form: with Ẽz = dz + µ z z dz + µ z 0 dt + µ z τ dτ + c z and Ẽz = dz + µ z z dz + µ z 0 dt + µ z τ dτ + c z .The ghost number 0 components of (B.1) lead to the 24 linear equations that have already been solved in Appendix [A] to compute the Beltrami spin connection ω(e).Its ghost number 1 components are to determine the explicit expression of the anticommuting Lorentz ghosts Ω ab (x) in terms of the Beltrami fields and the reparametrization ghosts.
The ghost number zero and form degree two component of this equation is which is the τ component of the vanishing torsion conditions of four-dimensional gravity solved in Appendix[A].Combined with the other ghost number zero components of T a = 0 it leads to the full system exposed and solved in Appendix[A], so one won't discuss the consequences of these ghost number zero equations in this appendix.
At ghost number one, T τ = 0 determines the transformation laws of M and a from its compo-nents proportional to dτ and dt as The terms proportional to dz and dz with ghost number one imply the following constraints for Ω τ z and Ω τ z Solving this system of two equations determines the Lorentz ghosts Ω τ z and Ω τ z , in terms of the Beltrami reparametrization ghosts, that have to be plugged back in (B.5), (B.6) and (B.9).The ω ab µ are the Beltrami field dependent expressions determined in Appendix[A].At ghost number two, T τ = 0 determines the transformation law of the reparametrization ghost c τ as At ghost number one, the terms proportional to dt and dτ provide Still at ghost number one, the terms proportional to dz and dz imply that Ω z0 and Ω 0z satisfy This system leads to the determination of Ω z0 and Ω 0z that have to be plugged back in (B.11), (B.12) and (B.15).Then, by combining (B.6) and (B.12), one can express Ω 0τ in terms of s(a + a).
The ghost number 2 components of the equation T 0 = 0 imply The terms at ghost number one and proportional to dz imply (B.17) Once combined with the analogous equation for sϕ implied by T z = 0, this equation will determine Ω zz as a function of the Beltrami fields and reparametrization ghosts, as well as the expressions of sϕ and sϕ.The term at ghost number one and proportional to dz is the analog of (B.17Let us now display the expressions of the 6 Lorentz ghosts that have been determined in functions of the Beltrami fields and the Beltrami reparametrization ghosts in this appendix, that is: + exp(ϕ) c z (ω τ z 0 + ω 0z τ ) − which are completely determined when a = −a and ϕ = ϕ = Φ 2 , just like in the Beltrami parametrization.In this particular case, which is the one useful for this article, the s transformations of the Beltrami fields can be fully determined by plugging the exact expression of the spin connection ω(e) (A.13), determined in Appendix[A], in the s transformations just derived in this appendix.