$\widehat{sl_2}$ symmetry of ${\mathbb R}^{1,3}$ gravity

We propose novel asymptotically locally flat boundary conditions for Einstein Gravity without cosmological constant in four dimensions that are consistent with the variational principle. They allow for complex solutions that are asymptotically diffeomorphic to flat space-times under complexified diffeomorphisms. We show that the resultant asymptotic symmetries are an extension of the Poincare algebra to a copy of Virasoro, a chiral $\mathfrak{sl}(2,{\mathbb C})$ current algebra along with two chiral $\mathfrak{u}(1)$ currents. We posit that these bulk symmetries are direct analogues of the recently discovered chiral algebra symmetries of gravitational scattering amplitudes as celestial CFT correlation functions.

The holographic description of any 4-dimensional gravitational theory with asymptotically locally flat boundary conditions is one of the most interesting questions being pursued in recent times.At the future or the past null infinity the boundary is a three dimensional manifold with a degenerate metric and the holographic theory is expected to include sectors residing at these null infinities.With Dirichlet type boundary conditions where the non-degenerate part of the boundary metric is taken to be that of a round S 2 the asymptotic symmetry algebra is the famous bms 4 [1][2][3] algebra that includes the Poincare algebra as its maximal finite-dimensional sub-algebra.In fact allowing for singular vector fields on S 2 extends the bms 4 to the so-called extended bms 4 algebra where the Lorentz transformation algebra sl 2 ⊕ sl 2 becomes two copies of Virasoro algebra [4][5][6][7].A different set of boundary conditions considered in [8,9] produced a different extension called generalised bms 4 that extends sl 2 ⊕ sl 2 to all smooth diffeomorphisms of S 2 .These two extensions have been shown [8,11] to be responsible for the sub-leading soft graviton theorems [12].However, it is fair to say it is unclear what the maximal set of symmetries admitted by R 1,3 gravity is.There is another and more urgent motivation to consider this question.Recently in [13,14] a novel sl 2 current algebra symmetry has been uncovered as the symmetry consistent with the OPEs of graviton operators in the celestial CFT.How such a symmetry algebra can be seen directly from classical gravity is an important question that we address in this paper.In particular, we seek boundary conditions for the R 1,3 gravity whose asymptotic symmetry algebra includes the sl 2 current algebra of [13,14].
Our analysis is for Locally Flat solutions in R 1,3 gravity, where motivated by chiral gravity boundary conditions for AdS 3 -gravity considered in [15,16], we allow for chiral fluctuations of the metric on the spatial part (Σ 2 ) of the null infinity (celestial sphere or celestial plane).These are non-radiative solutions on which we further impose variational principle with appropriate boundary action akin to AdS 3 -gravity case.The corresponding asymptotic symmetry analysis of such locally flat solutions leads to a novel extension of the Poincare algebra to an infinite dimensional algebra that consists of one copy of the Virasoro, one copy of the sl 2 current algebra and two u(1) current algebras which match with the symmetry algebra uncovered from the celestial CFT in [13,14].The resultant non-linear solutions we obtain after imposing the variational principle are characterised by Goldstone modes associated with the spontaneous breaking of this symmetry algebra in the gravitational vacuum.The results of our analysis include, • Providing all locally flat solutions (including complex ones) with time-independent boundary metrics on Σ 2 .
• Using a set of appropriate boundary terms extracting further conditions on the space of locally flat solutions coming from the variational principle.
• Choosing a chiral gauge for the metric on Σ 2 showing that the residual large diffeomorphisms generate the desired infinite dimensional chiral algebra that includes an sl 2 current algebra.
The rest of the paper is organised as follows.In Section 2 we review the relevant aspects of the two directions that have led us to our investigation, namely, the hidden symmetries 2d gravity of Polyakov from the AdS 3 perspective, and the recently uncovered chiral current algebra symmetries of the celestial CFT.In Section 3 we consider construction of asymptotically locally flat and exactly locally flat solutions of interest.In Section 4 the variational problem arising from the addition of the Gibbons-Hawking like boundary terms to R 1,3 gravity with a cut-off near future (past) null infinity is analysed.In Section 5 we carry out the asymptotic symmetry analysis of the solution spaces that solve the variational problem and compute the desired algebras.We provide a discussion of our results and open questions in Section 6.

A motivation and a review
In this section we review two aspects that act as motivation and guidance for the rest of our paper: The emergence of (i) sl 2 current algebra as part of the symmetries of the 2d celestial CFT, (ii) sl 2 current algebra in AdS 3 gravity with chiral gravity boundary conditions.

Current Algebra from Celestial CFT
The S-matrix elements of R 1,3 gravity can be recast as 2d conformal correlators, called the celestial amplitudes [19,20].By taking conformal soft limits one can uncover Ward identities of various 2d conformal currents -referred to as the conformal soft theorems.Here we review briefly the existence of two current algebra symmetries of the 2d celestial amplitudes that follow from the leading and subleading conformal soft theorems for positive helicity soft graviton operators (see [13] for notation and more details).For this one starts with the leading order conformal soft theorem for an outgoing positive helicity soft graviton operator S + 0 (z, z) where P k φ h i , hi (z i , zi ) = δ ki φ h i +1/2, hi +1/2 (z i , zi ), ǫ k = ±1 for an outgoing (incoming) particle.Also ) where ∆ k and σ k are the scaling dimension and helicity of the k-th particle respectively.The RHS of equation (2.1) is a polynomial in z, so we can expand it around z = 0 and rewrite the equation as Now if we define two currents C 1 2 (z) and C − 1 2 (z) in the following way, then equation (2.2) implies separate Ward identities of these two currents given by, and One can see from (2.4) and (2.5) that C 1 2 (z) and C − 1 2 (z) are generators of infinitesimal supertranslations acting as: respectively.Next we turn to the emergence of sl 2 current algebra from the celestial CFT.For this one starts with the corresponding subleading conformal soft graviton theorem (the holographic analog of [12]) for a positive helicity outgoing soft graviton operator S + 1 (z, z) where S + 1 (z, z) is given by One follows the same procedure as leading conformal soft theorem above.Expanding the RHS of (2.8) in powers of z gives Using this one can define three currents J i (z) where i = 0, ±1, which are the generators of sl 2 .In terms of these currents we can write the soft graviton operator S + 1 (z, z) as, The mode algebra of the currents J i (z) is 12) The symmetry algebra we have discussed so far has been obtained by analysing the correlation functions between the primary operators of the 2d celestial field theory and conformally soft gravitons.If a holographic duality exists between gravitational theories in four dimensional asymptotically flat spacetime and 2d celestial conformal field theory then the bulk theory should also posses these symmetries.This motivates us to search for these symmetries directly in the bulk R 1,3 gravity.
Next we turn to an older computation [15] in the context of AdS 3 gravity that led to a similar sl 2 current algebra.We will use an almost identical computation in R 1,3 gravity to uncover the desired symmetry algebra.

An sl 2 algebra from AdS 3 gravity
An sl 2 current algebra can be seen as the asymptotic symmetry algebra AdS 3 gravity by taking the boundary metric of asymptotically AdS 3 geometries in the Polyakov gauge [21].Working in the Fefferman-Graham gauge the locally AdS 3 geometries can be written as db , ∇ a (0) g Choosing the chiral gravity gauge of Polyakov for the boundary metric g ab the differential conditions impose the following equation: One also has Finally the bulk variational problem can be satisfied by setting g z z = 0 as shown in [15].This makes F (z, z) a polynomial of degree 2 in z.The residual bulk diffeomorphisms lead to one copy of sl(2, R) current algebra and one copy of Virasoro algebra. 1n what follows, we implement a similar procedure in R 1,3 gravity and demonstrate that an analogous sl 2 chiral current algebra emerges in this context as well.

A class of ALF spacetimes in four dimensions
We start with collecting some useful formulae towards construction of asymptotically locally flat (ALF) solutions in R 1,3 gravity.We find it convenient to use the Newman-Unti gauge [22] and work with the coordinates (r, u, z, z) suitable for the future null infinity.Then we have g rr = g rz = g rz = 0, g ur = −1. (3.1) Let us write the remaining metric components g ij for i, j ∈ {u, z, z} in the following form: We seek Ricci flat metrics (R µν = 0) which are also asymptotically locally flat.The latter condition is implemented by demanding that R µ νσλ → 0 as r → ∞ [23].We first solve for R µν = 0 for large r in a power series in 1/r.The leading non-trivial conditions require that g (0) ij is degenerate.Since we are interested in solutions for which the metric g (0) ab for a, b ∈ {z, z} on the spatial manifold Σ 2 is non-degenerate we solve this condition by assuming: Then the next non-trivial condition implies that g (1) ab ).One also finds the condition det (γ) = 1 4 (tr γ) 2 , for the cb .Any 2 × 2 matrix that satisfies such relation has to have both its eigenvalues the same -and therefore can only be proportional to the 2 × 2 identity matrix I 2 .Thus we arrive at the condition: ab ∀a, b ∈ {z, z} with the same λ.This immediately leads to g (0) ab (u, z, z) = Ω(u, z, z)q ab (z, z) where q ab is u independent general 2 × 2 matrix.At O(1/r) from vanishing of R uz and R uz we find g uz = 0. We will further assume that the conformal factor Ω for the boundary metric g ab is independent of u which in turn implies g (1) uu = 0. Further one finds: ab (u, z, z) = D ab (z, z) + 1 4 g (1)  ac g cd (0) g cb .
(3.6) and so on, where V a (z, z) and D ab (z, z) are unconstrained u-independent 2d vector and rank-2 symmetric tensor respectively.It turns out that imposing vanishing of R µ νσλ at order r implies ab − This means that one can determine the u-dependence of the trace-free part of g ab completely and it is at most linear in u.Any such solution can be written as where {g ab (z, z)} are traceless and symmetric.The remaining data in (g ab ) has to satisfy some further differential conditions.In particular: where R (0) is the scalar curvature of the 2d metric g ab .This equation (3.9) is similar to that in (2.14) of AdS 3 gravity.
Further constraints arising from asymptotic flatness are: V a (z, z) = 0, ∂ u g uu determined in terms of the data at order r 2 and order r and differential conditions on D ab .More unconstrained data also appears at O(r −1 ) and Ricci flat solutions can be systematically constructed order by order in powers of r −1 .It turns out that g ab (0) g ab remains unconstrained and we set it to zero as a further gauge condition [22].
However, to study the asymptotic symmetries of these solutions one does not need to find these solution spaces completely.It is sufficient to find the locally flat (LF) solutions that share the boundary conditions derived so far.

Classes of locally flat geometries
These form a subset of Ricci flat geometries discussed above which have vanishing Riemann tensors.With our NU gauge and boundary conditions we find that any such locally flat solution can be written as polynomials in r: ab + rg (1) ua dx a du (3.10) where ba , g db , g ab (0) g Along with the differential conditions: db − g When these differential conditions (3.13, 3.14) are solved for g (0) ab and g ab then the rest of the 4d metric components (g ua , g ab ) can be found using (3.11, 3.12).The u-dependent part of the equation (3.14) is satisfied identically when we use (3.13) and thus can be simplified to:

Complete set of solutions
It turns out that the equations (3.13, 3.14) can be solved completely which we turn to now.To solve these equations first we take the boundary metric g ab in the form where (f , f , Ω) are arbitrary functions of (z, z).Then we further parametrise f (z, z) and f (z, z) as follows in terms of two other independent functions ζ(z, z) and ζ(z, z): Notice that this parametrisation is not one-to-one: suppose Then the solutions for g ab components can be given explicitly in terms of the functions (f, f , Ω) and ρ = ∂ζ ∂ ζ −1/2 and their derivatives.
To keep the expressions simple we drop all the coordinate dependences (z, z) of various functions and use where a ∈ {z, z}.Then components g (1,0) zz and g (1,0) zz are given by: where υ = υ(z, z) is also arbitrary.Next, the g (1,1) zz is given by The components g (1,0) z z and g (1,1) z z can be obtained from g (1,0) zz and g (1,1) zz respectively by f ↔ f and ∂ → ∂ (with ρ, υ and Ω unchanged). 2  To summarise, a general locally flat metric solving equations (3.13, 3.14) is parametrised by the following arbitrary set of functions: ab and ρ, and υ(z, z) in g (1) ab . 3The remaining components g (1,1) z z and g (1,0) z z can be obtained from the above ones using the tracelessness condition of g (1) ab : zz f + g z z f . (3.22)

Gauge fixing the boundary metric
As shown in the appendix A the residual diffeomorphisms that preserve the NU gauge and the boundary conditions act as diffeomorphisms and Weyl transformations on g (0) ab .These boundary diffeomorphisms can be used to gauge fix the 2d metric g (0) ab down to one independent function.The most commonly used gauge choice for 2-dimensional metrics is the conformal gauge.In our language this amounts to setting f (z, z) = f (z, z) = 0 and letting Ω fluctuate.However, as we are interested in getting a symmetry algebra that includes sl 2 current algebra we choose the analogue of Polyakov Gauge : f (z, z) = 0 or f (z, z) = 0 with Ω fixed to a given function. ( Below we provide the solutions in this gauge.

Locally flat solutions in Polyakov gauge
To be specific, we choose f = 0. Then the components of g ab reduce to zz = 0 (3.24) Similarly the components g (1,0) ab and g (1,1) ab in (3.19, 3.20) reduce to: Under this change the resulting configurations remain LF solutions. 3We have generated this solution by starting with flat spacetime with metric ds 2 R 1,3 = −2 du dr + r 2 dz dz and making a finite coordinate transformation that keeps us in the chosen gauge.A similar exercise was done in [17,18] with a different parametrisation of the boundary metric.We find our parametrisation better suited for the problem at hand.The solution in [18] matches with our LF solution after the following identifications, Φ = log ρ(z,z) −2 Ω(z,z) . We also have to take Now that we know g ab and g ab the rest of the components can be found using the relations (3.11, 3.12).We will not be interested in general conformal factor Ω(z, z) in what follows -but only two choices Ω = 1 and Ω = 4 (1 + zz) −2 .These choices have the property that the quantities τ zz , τ z z vanish.

Restriction to g
(1) zz = 0 In the next section we will show that the solutions with vanishing g (1,1) zz and g (1,0) zz also satisfy the variational principle.Anticipating that result we write down the solutions with g (1) zz = 0.In the cases for Ω's with τ zz = τ z z = 0, we see from (3.25) that ) turns out to be a polynomial of degree two in z: where det g := g 1 (z) g 4 (z)−g 2 (z) g 3 (z).For (3.30) to make sense we have to assume that det g = 0. 4 Furthermore, where We can equivalently write g Now we will leave the Polyakov gauge solutions 5 found in this section in store until we learn to impose the variational problem -which will be done in the next section. 4There is another interesting interpretation of the coefficients J a of the powers of z in (3.30) as follows.First consider the 2 × 2 matrix which is an element of the algebra gl 2 .Consider a 2 × 2 matrix representation of sl 2 algebra [t a , t b ] = (a − b) t a+b -for instance, in terms of the Pauli matrices we can take where the non-zero components of η ij are η +1 −1 = η −1 +1 = 1/2 and η 00 = −1.One also has Tr [A(z)] = 1 2 det g ∂ z det g and without loss of generality we may choose det g = 1.This makes A(z) ∈ sl 2 and A(z) = J a (z) t a . 5Given these Polyakov gauge solution we can readily find the solutions in conformal gauge.We simply have to set f (z, z) = 0 which can be done by choosing ζ(z, z) = ζ(z).

Boundary terms and variational principle
We now turn to the variational problem among the configurations we have considered so far.To be precise we consider our configuration space to be all the 4-dimensional metrics g µν that are in the NU gauge which asymptotically approach the locally flat solutions.Then we will define our solution space to be a subset of Ricci flat configurations, and equally importantly, that also satisfy a variational principle δS = 0.
The usual prescription of boundary terms consists of adding a Gibbons-Hawking like term and a set of possible counter terms to the standard Einstein Hilbert action S EH .There are two essential aspects required of such boundary terms: 1.The boundary action S bdy has to be consistent with the symmetries that leave the gauge choice and the boundary surface invariant.
2. The variation of S EH + S bdy should be proportional to variations of metric data on the boundary -but not its derivatives.
The standard Gibbons-Hawking term is actually invariant under the full set of 3d diffeomorphisms of the boundary.In the context of AdS n+1 gravity in the Fefferman-Graham gauge residual symmetries are the diffeomorphisms of the n-dimensional subspace r = r 0 and thus adding the Gibbons-Hawking term and other counter terms that are also invariant under the n-dimensional diffeos is justified.Now we pose this question in our context: what are the subset of asymptotic symmetries of the class of geometries that leave r = r 0 fixed.Generators of any such coordinate transformation has to have ξ r = 0.These vectors of course have to continue to solve (A.1, A.3, A.4).The first consequence of this condition ∂ r ξ r = 0, is that (∂ u − g ua g ab ∂ b ) ξ u = 0. Since ξ r (0) = ξ u (1) = ∂ u ξ u we also have ∂ u ξ u = 0 implying g ua g ab ∂ b ξ u = 0. From (A.4) working in the gauge g (0) ab g ab (1) = 0 we see that ξ r (1) = 0 requires ξ u to be a harmonic function in 2d.Since the harmonic equation is Weyl invariant and in 2d every metric is Weyl equivalent to flat space the solutions to This equation is background dependent and a linear combination of a holomorphic and anti-holomorphic functions ) and unless the coefficients (g u z , g u z ) are zero (and non-generic) the only solutions are and this is what we work with.To summarise the subset of symmetries of our class of geometries that leave r = r 0 invariant are: -that is, rigid translations in u and arbitrary diffeomorphisms of (z, z).
Another way to arrive at the same conclusion is the following.We should have expected that the residual transformations do not mix different orders of powers of r.This in turn implies that ξ a is independent of r (ξ u is already independent of r).Then the last of (A.1) implies ∂ a ξ u is zero, and the fact that ξ r should also vanish requires ∂ u ξ u = 0, thus making ξ u a constant and ξ a = ξ a (0) (z, z).These are the generators of the boundary coordinate transformations: The Jacobian of such transformations is: ∂u ′ ∂u = 1, ∂u ′ ∂x a = ∂x ′a ∂u = 0 with the only non-trivial part J a b = ∂x ′a ∂x b .Now we seek the boundary terms that respect at least these symmetries.
The bulk four-dimensional metrics in the NU gauge are of the form: where g ab is the inverse of g ab , g u a = g ub g ba , etc.We take the unit normal to the r = r 0 surface as: The induced metric on r = r 0 surface is: with its inverse with N 2 = −g uu + g au g a u and the usual completeness relation g µν = n µ n ν + γ ij e µ i e ν j .Thus the metric data in the line element on the boundary is (g uu , g ua , g ab ).The expected set of symmetries of the boundary (u → u ′ = u + u 0 , x a → x ′ a = x ′a (x)) is much smaller than the 3d diffeos in (u, x a ).Therefore, we expect much more freedom in the choice of the boundary terms.The transformations of the components of the induced metric under the reduced symmetry are: This enables us to classify the basic scalars under the boundary symmetries and some of them are: no derivatives : g uu , g ab g ua g ub , one derivative : The integration measure invariant under our boundary symmetry (u, x a ) → (u + u 0 , x ′a (x)) is du d 2 x √ σ where σ = det g ab , which can be used to integrate any function of the scalars listed above to obtain a potential boundary term.

The boundary terms
Now we look for the potential Gibbons-Hawking type terms we can construct consistent with our boundary symmetries.The bulk action is the standard Einstein-Hilbert action Then the variation of the action within our configuration space around a configuration satisfying the Einstein equation R µν = 0 is where J κ = g µν δΓ κ µν − g µκ δΓ ν µν where δΓ κ µν = 1 2 g κω (∇ µ δg ων + ∇ ν δg µω − ∇ ω δg µν ) which leads to J µ = g µν g σλ (∇ σ δg νλ − ∇ ν δg σλ ).We will have to manipulate this term carefully.Since our surface is defined by r = r 0 the unit normal is n µ = Nδ r µ where N = 1 √ g rr .But we also have g rr = γ g where γ = det(γ ij ) and g = det g µν =⇒ √ −γ = N √ −g.This boundary term can be written explicitly for geometries in the NU gauge with n µ = 1 √ g rr δ r µ and we find: where we introduced the notation: ω = g uu , v a = g ua and σ ab = g ab along with σ ab being the inverse of σ ab and v a = σ ab v b .In this we seek that those terms with tangential derivatives (that is, derivatives w.r.t (u, z, z)) on variations of the boundary data (δω, δv a , δσ ab ) have to be cancelled.The usual strategy involves completing such terms into either total variations or total derivatives so that the variations (derivatives) are moved away from the derivatives (variations).To do this one may use the following identities: Using these and moving the total variations and total derivatives (in (z, z) coordinates) to the lhs we find: Note that there are no terms involving tangential derivative of (δω, δv a , δσ ab ) on the rhs.Thus the boundary action required to be added to the Einstein-Hilbert action (4.7) in our context is, and the total divergence term in δS EH is (16πG) −1 times: that we ignore (this amounts to assuming that the geometry of Σ 2 with coordinates (z, z) is either compact or, when it is not, the integrand falls-off fast enough near its asymptotes).In summary we have managed to find the boundary action (4.11) we sought such that the variation of the total (bulk plus boundary) action is a linear combination of (δω, δv a , δσ ab ) but not their derivatives: where t ab = σ ac σ bd t cd , j a = σ ab j b .Finally we can expand the integrand δL EH+bdy around large-r, and we find: uu −r g (0) 2 δg (2)  uu + g (2)  uu g ab (0) δg where: Tr g (1) Tr g (2) − Tr g (1) g −1 (0) g (2) g −1 Tr g (1) Tr g (1) g −1 (0) g (1) g −1 Tr g (1) g −1 Tr g (1) ∂ u g (1) ab − ∂ u (Tr g (1) )g ab δg ab (0) − Tr (∂ u g (2) ) g ab δg ab (0) +g (2)  ua g ab (1) δg bu + g (2)  uu g ab (0) δg with ab and so on.As we will be taking the boundary r = r 0 → ∞ we will not need to keep terms that vanish in this limit.

Conditions from variational principle
We first impose for all configurations as these are part of the boundary conditions derived in Section 3. Furthermore, since we consider the class of geometries that are locally flat we restrict to the class (3.10) of geometries: g We also impose the gauge conditions and Tr g (1) = 0 as well.Then we have ab δg ab (0) − Tr (∂ u g (2) ) g ab δg ab (0) + g (2)  uu g ab (0) δg We just have to ensure that this quantity δL, when integrated over the boundary directions (u, z, z), vanishes in the r = r 0 → ∞.One more condition that we will impose for all the cases below is: which when using the relation g uu = − 1 2 R 0 is equivalent to holding the Euler Character of the metric on Σ 2 fixed under variations (see also [18]).Now we consider special cases of boundary conditions that ensure δS = 0.
To summarise the variational principle in the Polyakov gauge can be solved by imposing: on the class of locally flat geometries we found in the previous section.
In anticipation of this result we have already presented the solutions satisfying these additional conditions in section 3.3.2.Here we present the locally flat solutions in closed form for two cases: Ω = 1 and Ω = 4 (1 + zz) −2 that also are consistent with our variational principle.
• Locally flat solutions Ω = 1: • Locally flat solutions with Ω = 4 (1 + zz) −2 : Here all the functions (κ, J i , C r ) are holomorphic and the non-zero components of η ij are η −1 +1 = 1/2 and η 00 = −1, and ǫ − 1 2 + 1 2 = 1.Now that we have obtained the locally flat solutions with the boundary conditions we wanted as well as imposed a consistent variational problem we can turn to analysing the symmetry algebras of these solution spaces.

Asymptotic symmetries
Here we seek all the vector fields that keep us within the classes of solutions (4.23, 4.24) and compute their commutator algebras.

The vector fields
From the general analysis of asymptotic symmetries done in (A.1-A.3) the first few components of the vector fields that keep us within the class of (4. 23, 4.24) are given by, (5.4) In the above solution V a = (Y (z), Y (z, z)).Under the action of these vector fields the data g (0) ab and g (1 ab transform as follows, ab + ξ u ∂ u g (1) We now have to impose conditions coming from the variational principle: g z z = g This requires setting D z D z ξ u (0) = 0 and D zD z D a V a = 0 separately.
• The equation D z D z ξ u (0) = 0 can be solved for ξ u (0) and we obtain • The equation D z D z D a V a = 0 allows us to solve for Y (z, z) as in either case of Σ 2 .
Therefore, the vector fields that keep us within our solution spaces are characterised by six arbitrary holomorphic functions: . The subleading components of vector fields are then found in terms of these leading order components as mentioned in the general analysis done in Appendix A. If these holomorphic functions are allowed to have poles, then the supertranslation vector field ξ u (0) for a specific choice of P − 1 2 , P 1 2 as follows, The vector field in equation (5.10) was used in [10] to show the equivalence between supertranslation Ward identity and Weinberg leading soft theorem for positive helicity graviton localised at the reference direction (w, w) on the sphere.The vector field in (5.10) induces transformation of g (1;0) zz which is zero everywhere on the sphere except singular in the direction (w, w), δg In a similar spirit for the vector field in equation (5.9) the choice, (5.12) coincides with the vector field used in [9] to show the equivalence between Diff(S 2 ) Ward identity and Cazchao Strominger subleading soft theorem for positive helicity graviton (the conjugate vector field give rise to Ward identity that is equivalent to CS soft theorem for negative helicity graviton) 6 .For this particular choice of vector field, δg The singularity in (5.11) and (5.13) is similar in nature to the one generated by superrotation vector fields of the type which change the sphere metric according to (5.5) at null infinity by adding singularities at isolated points [24].

Algebra of vector fields
Now that we have our full set of vector fields that enable us to move in the space of solutions (4.23, 4.24), one can check that the Lie brackets between any two of these vector fields close under the modified commutator (see for instance [4]) Calculating the bracket as in eq (5.15) we find: The field κ(z) transforms like a quasi primary of weight (h = 2) with an inhomogeneous term that is similar to the chiral stress tensor component of a 2d CFT.The C r (z) are related to supertranslation current and transform like the primary of weight (h = − 1 2 ).The fields κ(z), J i (z), C r (z) also transform as Kac-Moody primaries with j = 0, 1, 1  2 respectively under the sl 2 algebra.As we saw in section 3.3 after imposing variational problem g z z = 0 the locally flat solutions are given in terms of fields, with the condition g 1 (z) g 4 (z) − g 3 (z)g 2 (z) = 1.This condition may be used to eliminate g 1 (z) (whenever it is non-zero) in terms of other three fields.These fields are more fundamental as (κ(z), J i (z), C r (z)) are defined in term of these fields (see 3.30, 3.35).We can find the transformations of these more fundamental fields under the action of the vector fields specified by (Y (z), Y i (z), P r (z)) and we obtain: Therefore, the fields g 2 (z), g 3 (z), g 4 (z) transform like scalars under left Virasaro transformation Y (z).To write the transformation properties of υ (1) and υ (2) we define: Then we find that υ(1) (z), υ(2) (z) transform as: (5.29)

Comments on the charges
The locally flat solutions that we have obtained parametrise the space of gravitational vacua.The fields {κ(z), J i (z), C r (z)} can be thought of as 'Goldstone' modes associated with spontaneous symmetry breaking of Virasaro, chiral sl 2 current and u(1) current algebra symmetry.In order to construct complete phase space we are required to construct soft charges that generate symmetry transformation on these fields.From what we have seen the fields in the solutions actually behave like (quasi-) primaries under the chiral Virasoro generators as well as the current algebra ones and thus one expect the charges to be given more appropriately as line integrals as it is the norm for a chiral conformal field theory.It is not clear how to obtain such line integral charges from the 4d bulk perspective where the usual route of defining charges through covariant phase space formalism [27] or Barnich-Brandt method [28] lead to co-dimension two surface charges that are defined on the celestial sphere S 2 .
In [18] charges for gravitational vacua exhibiting supertranslation and Diff (S 2 ) symmetry were constructed and it was shown that these co-dimension two charges corresponding to supertranslations vanish but charges for superrotations which are proportional to terms quadratic in g (1,0) ab and their derivatives are in general non-vanishing and conserved.These charges evaluate to zero for our chiral solutions in (4.23, 4.24).Therefore one needs to find an alternative prescription to calculate charges that are line integrals and provide representation of chiral sl 2 algebra and generate correct symmetry transformation on solution space.
In an ongoing work which we will report in the near future we find that to get the charges as line integrals one can switch to the first order formalism of gravity where these locally flat solutions can be written as connections A µ that takes value in iso(1, 3) algebra.The condition of vanishing curvature for this connection (F µν = 0) is equivalent to local flatness condition that we used to solve in the 4d Einstein gravity.One can gauge away the r dependence using gauge transformation and get an effective 3d flat connection.The sector within this 3d gauge connection parametrised by fields {κ(z), J i (z)} find a natural interpretation independent of field C r (z) in terms of 3d flat sl(2, C) Chern-Simons connection.The residual gauge transformations that preserve the form of this connection are a subset of the asymptotic symmetries found in previous section (the sl 2 algebra and the Witt algebra). 7he charges generating residual gauge transformations in a Chern-Simons theory are line integral given by, where Λ is the gauge parameter and δA = dΛ + [Λ, A].Using the formula (5.30) the charges generating chiral sl 2 current and Virasaro symmetry are given by The part of gauge connection that has the information about the mode C r (z) associated with supertranslation is gauged by the momentum generators (P i ) of Poincare algebra and if one attempts to calculate charges associated with supertranslation using (5.30) one has to consider full iso(1, 3) gauge connection as δA µ gets non-trivial contribution from the commutators of sl(2, C) generators with P i .Since Poincare algebra does not have a non-degenerate bilinear invariant, therefore computation of charges for u(1) currents is not possible using the charge formula (5.30).
Another possible way to obtain these supertranslation charges could be to start with the so(2, 3) algebra which does have a non-degenerate bilinear invariant.The analogous flat connections in this case describe locally AdS 4 solutions having the chiral fields {κ(z), J i (z), C r (z)}.These solutions in the flat space limit, i.e., when the radius of AdS 4 goes to infinity, match with the corresponding locally flat solutions.After obtaining charges as line integrals one can write down a set of holomorphic operator product expansions between the charge generators and the fields.We hope to get some insight into the supertranslation charges in the R 1,3 context via a suitable Inonu-Wigner contraction of this computation.This will also allow us to find the central extensions (if any) of the algebra of charges that cannot be seen from the commutator algebra of the corresponding vector fields.This work is currently in progress.

Conclusion and Discussions
In this paper we introduced a new set of boundary conditions for the four dimensional classical gravity without cosmological constant (the R 1,3 or R 2,2 gravity) that are asymptotically locally flat near the null infinities.Our boundary conditions are non-Dirichlet.By considering a Chiral gauge for the metric on 2-dimensional spatial manifold Σ 2 motivated by Polyakov's gauge for asymptotically locally AdS 3 of [15] we provide the complete class of locally flat solutions.Constructing and making use of a set of appropriate boundary terms to impose a variational principle, we restrict the class of locally flat solutions to a subset.By studying the symmetry algebra of the vector fields that leave our final solution space invariant, we show that it is a novel extension of the Poincare algebra that includes a chiral sl 2 current algebra.Our symmetry algebra is identical to the recently uncovered symmetry algebra of the celestial CFT from its conformal soft theorems [13,14].
Our variational problem is defined for 4d Einstein gravity with a specific set of boundary terms we proposed in the text.To ensure δS = 0 for the allowed classical solutions we have chosen to impose additional boundary conditions on the configurations that already solve the bulk equations of motion. 8We would like to emphasise that even in the case of Dirichlet boundary conditions (in which the metric on Σ 2 is held fixed) there are additional conditions imposed by our variational principle and solving them in a non-chiral fashion requires us to set g (1,1) zz = g (1,1) z z = 0.Even though we did not provide the details it can be seen easily that the residual large diffeomorphisms of such Dirichlet class of solutions does not permit extension of the Lorentz algebra part of the bms 4 into two copies of Witt algebra.Thus our boundary conditions would not allow the extended bms 4 as the asymptotic symmetry algebra in the Dirichlet case.It will be important to explore the consequences of this fact further.
We provided exact solutions that are locally flat spacetimes.Working in the NU gauge these are polynomials in radial coordinate r.When the boundary metric is taken in the Chiral Polyakov gauge, the solution spaces are necessarily complex.Starting from our solution space we can generate three more classes of geometries by T : u ↔ −v and/or P : z ↔ −1/z -which are time-reversal and spatial parity transformations respectively in the asymptotic flat spacetime (with Σ 2 = S 2 ).We would like to think of these four classes of solutions as valid in four different coordinate patches of the fully extended (locally flat) geometries.Then these four patches are related to each other by the (P, T) transformations.One can in principle patch together these solutions that are (C) PT invariant.Just as there are four classes of solutions there are four corresponding chiral algebras.This is similar to the two copies of bms 4 symmetry algebras in the context of Dirichlet boundary conditions -one at the past null infinity and the other at the future null infinity.Just as an appropriate combination was found by Strominger (see [30] for a review) picked by the CPT invariance of the scattering amplitudes we expect that an appropriate combination of the four copies of our current algebras as well to emerge as the correct symmetry of the scattering amplitudes.
The locally flat solutions (4.23) and (4.24) that we arrived at (and worked with) after imposing chiral boundary conditions are indeed complex in the R 1,3 gravity (they are real however in the R 2,2 gravity as (z, z) will be light-cone coordinates (x + , x − )) 9 .As a consequence the vector fields generating the asymptotic symmetries of this class of geometries are also complex.Since for LF solutions the Riemann tensor vanishes, in principle these solutions can all be obtained (in open patches) by finite complex coordinate transformation of metric η µν of the Minkowski spacetime R 1,3 .Expanding the LF solutions around η µν to the first order in the six holomorphic functions the perturbation h µν can be written as pure diffeomorphisms with the same complex vector fields that generate our asymptotic symmetries.In [35] it was shown that the conformal basis for graviton wave-functions h ∆,± µν;a of definite helicity (where ± superscript denotes whether the graviton is incoming/out-going and the subscript a denotes its helicity) in the soft limits ∆ → 1, ∆ → 0 become pure diffeomorphisms: h . Near null infinity these vector fields ζ ∆ µ;± in the soft limit become generator of supertranslation symmetry and Diff(S 2 ) symmetry and for positive helicity soft graviton take the form of (5.10), (5.12) respectively .The conformal primary wave functions given in terms of these vector fields are then interpreted as Goldstone modes of spontaneously broken asymptotic symmetries of gravity theory 11 .Similarly one can interpret our complex solutions as the 'condensates' of soft gravitons of definite helicity-positive (negative) if the solutions are characterised by six holomorphic(anti-holomorphic) functions.
In this paper, we use the definition of variational principle in the conventional sense that implies stationarity of the action S EH+bdry. on the solutions.As argued in [31], one modifies the definition of a well-defined variational principle in the presence of radiation to allow for some presymplectic flux through the boundary.However, we believe that even if we consider radiative solutions, then in the asymptotic regions unaffected by the radiation (for instance, if a pulse of radiation reaches the boundary during some finite interval in u, then sufficiently far away from this interval), the solutions should approach one of the vacuum solutions.And these vacuum solutions, we posit, have to be one of the classes of locally flat solutions for which the total action is stationary.It will be interesting to see whether our symmetry analysis still holds in the case of presymplectic flux, through the boundary, instead of the stationarity condition of the action.
A bigger symmetry algebra (w 1+∞ ) than the one we uncovered in this paper is observed from the conformal soft limits of graviton operators in the celestial CFT [32,33] recently.It will be interesting to explore if our boundary conditions can be generalised to incorporate these extended symmetries or not. 12Following the analysis of [36,37], one expects that the vector fields generating w 1+∞ are overleading in r as one approaches the boundary of spacetime.However, it can be checked from eq. (A.2) that such behaviour for vector fields in the NU gauge does not exist.Therefore, to extend the analysis of algebra in our present work to w 1+∞ algebra, one has to solve for vector fields in a different gauge.In [35], the authors derived the vector fields that generate arbitrary diffeomorphisms of the celestial sphere in harmonic gauge.One can do a similar analysis as theirs by demanding less restrictive boundary conditions that allow for metric perturbation around Minkowski spacetime to fall off as O(r n ) for n ≥ 2 and obtain vector fields that are overleading in r near null infinity.It is plausible that imposing a variational problem for such configuration might further restrict the solution space, the residual gauge transformations of which could map to w 1+∞ .However, the interpretation of solutions with such overleading behaviour in r near the null infinity from the spacetime perspective is unclear as they do not obey the standard asymptotically flat boundary conditions.
We considered the metric on Σ 2 in the Polyakov gauge.The other natural choice is the conformal gauge in which we have found a class of locally flat solutions (see Appendix B for details).We are currently analysing the symmetry algebra of these solution spaces and what these would mean for the scattering amplitudes of gravitons.
The Chiral gravity boundary conditions of [15] that gave rise to the Polyakov's chiral sl 2 algebra from the AdS 3 gravity was generalised to the supersymmetric context in [39].It should be possible to generalise the boundary conditions of this paper to 4d gravitational theories with supersymmetry to find supersymmetric extensions of the Chiral sl 2 current algebra found here.Very recently such supersymmetric extensions have been observed in the celestial amplitudes [40].
In [41] 3-dimensional scalar field theories coupled to Carrollian geometries were constructed that were manifestly invariant under a set of Carroll diffeomorphisms and Weyl transformations.Gauge fixing those symmetries lead to field theories with bms 4 symmetries.It will be interesting to see if a gauge fixing similar to the one considered here leads to the emergence of sl 2 current algebra in such theories or not.
We have obtained our chiral algebra from the future null-infinity.The generalised bms 4 has been shown to emerge from the asymptotic symmetry analysis around the time-like infinity as well [42].It will be interesting to analyse our problem from the point of view of the time-like infinity.for which the curvature is r 0 = 2β/α 2 .Whenever r 0 > 0 we set α = β = 1 (which means we take r 0 = 2) and for r 0 < 0 we set α = −β = 1 (which corresponds to r 0 = −2).
Let us also note that as a consequence of the consistent variational problem we have δR 0 = 0.This still leaves two terms proportional to g ab (0) ∂ u g (2) ab and D a g (2) au .Since we impose g

11 )
and we just have to imposeD a g (2) au = D a D b gThis condition restricts the choice of C(z, z) in (3.10).