Aspects of the BMS/CFT correspondence

Transcription

1 DAMTP, Cambridge. February 17, 2010 Aspects of the BMS/CFT correspondence Glenn Barnich Physique théorique et mathématique Université Libre de Bruxelles & International Solvay Institutes So thank you for the occasion to speak here. Today I would like to talk about how 2 dimensional conformal field theory arises directly in the context of classical 4 dimensional GR. Phrased differently, I will discuss some gravitational aspects of the AdS/ CFT correspondence which remain when the cosmological radius goes to infinity.

2 Overview Classical gravitational aspects of AdS3/CFT2 correspondence 4d flat case, null infinity: asymptotic symmetries 3d flat case, null infinity: BMS3/CFT1 correspondence 4d flat case, null infinity: solution space work done in collaboration with C. Troessaert In order to motivate the results in the 4 dimensional case, I will start with gravity in 3 dimensions and quickly review some of the gravitational aspects of the ADS3/CFT2 correspondence. In this case, boundary conditions are given at spatial infinity. Then I will move to the 4d flat case and consider null infinity. I will show that the asymptotic symmetry algebra, the so-called BMS algebra as defined in the early 60 s, can be taken to involve besides the supertranslations a left and right copy of the Virasoro algebra in a completely natural way and will derive the first consequences of this observation. As a digression, I will then consider the 3 dimensional flat case to illustrate how the bms3 algebras acts on solution space and discuss the central extension in this context. Finally, I will return to the 4d case and its solution space and work out the transformation properties of Bondi s news tensor, mass and angular momentum aspects under local conformal transformations.

3 AdS3/CFT2 Asymptotic symmetries Fefferman-Graham ansatz g µν = l 2 r g AB g AB = r 2 γ AB (x C )+O(1) r t, φ 2d metric γ AB conformal to flat metric on the cylinder γ AB = e 2ϕ η AB η AB dx A dx B = dτ 2 + dφ 2, τ = t l, ϕ = ϕ(xa ) asymptotic symmetries L ξ g rr =0=L ξ g ra, L ξ g AB = O(1), general solution determined by conformal Killing vector ξ r = 1 2 ψr, ξ A = Y A + I A, I A = l2 2 Bψ r dr r Y A of η AB g AB = l2 4r 2 γ AB B ψ + O(r 4 ). ψ = D A Y A The Fefferman-Graham ansatz consists in defining an asymptotically AdS3 spacetime by a gauge fixed metric, where the dynamical variables are all encoded in a 2d metric whose leading order as one approaches spatial infinity is conformal to the flat metric on the cylinder. This class of spacetimes includes besides anti-de Sitter space also the BTZ black hole for instance. Asymptotic symmetries are defined as those infinitesimal transformations that leave the FG form invariant. The corresponding equations can be solved exactly to all orders. The generators are determined by a conformal Killing vector of the boundary metric and thus of the flat metric on the cylinder. The subleading orders of these vectors also depend explicitly on the metric. In some sense, these transformations constitute the residual global symmetries after the Fefferman-Graham gauge fixation.

4 AdS3/CFT2 Asymptotic symmetries metric dependence ξ µ = ξ µ (x, g) δ g ξ 1 g µν = L ξ1 g µν modified bracket [ξ 1, ξ 2 ] µ M =[ξ 1, ξ 2 ] µ δ g ξ 1 ξ µ 2 + δg ξ 2 ξ µ 1 faithful representation of conformal algebra [ξ 1, ξ 2 ] r M = 1 2 ψr, [ξ 1, ξ 2 ] A M = Y A + I A, Y A =[Y 1,Y 2 ] A, ψ = DA Y A light-cone coordinates x ± = τ ± φ, 2 ± = τ ± φ, γ ABdx A dx B = e 2ϕ dx + dx Y ± (x ± ) ± = n Z c n ±l ± n, l ± n ± = (x ± ) n+1 ±, [l ± m,l ± n ]=(m n)l ± m, [l ± m,l n ]=0 include Weyl rescalings of boundary metric L ξ g rr =0=L ξ g ra, L ξ g AB =2ωg AB + O(1) direct sum with abelian algebra of Weyl rescalings ( Y,ω) = [(Y 1, ω 1 ), (Y 2, ω 2 )] Y A = Y B 1 B Y A 2 Y B 2 B Y A 1, ω =0 If one then introduces the following modified bracket, which takes into account the metric dependence of the asymptotic Killing vectors, one finds that the latter form a faithful representation of the conformal algebra. In this case here, the spacetime vectors form a faithful representation of the conformal algebra in the usual bracket to leading order and the modification is not so essential, but in the cases to be discussed below, this is no longer necessarily the case. For later use, let me also shortly flash the explicit form of the conformal algebra in lightcone coordinates, where we have two copies of the Witt algebra. One can also generalize the above considerations by including the conformal rescalings of the boundary metric, or in other words, shifts in phi. The result is then the direct sum of the conformal algebra with the abelian algebra of Weyl rescalings. My point here is that both transformations coexist and really play a different role as I will now show. The same extensions by conformal rescalings exists in the flat case to be disucssed later, but for simplicity, I will not mention it again in that case.

5 AdS3/CFT2 Solution space existence of general solution integration constants Ξ ++ = Ξ ++ (x + ), Ξ = Ξ (x ) when ϕ =0 g AB dx A dx B = (r 2 + l4 r 2 Ξ ++Ξ )dx + dx + l 2 Ξ ++ (dx + ) 2 + l 2 Ξ (dx ) 2, BTZ black hole Ξ ±± =2G(M ± J l ) ADS3 general solution g AB dx A dx B = e 2ϕ r 2 +2γ + r 2 e 2ϕ (γ γ ++ γ ) dx + dx + +γ ++ (1 r 2 e 2ϕ γ + )(dx + ) 2 + γ (1 r 2 e 2ϕ γ + )(dx ) 2, γ ±± = l 2 Ξ ±± (x ± )+ 2 ±ϕ ( ± ϕ) 2 γ + = l 2 + ϕ Using the FG form of the metric, one can solve the 3d Einstein equations with negative cosmological constant exactly. The general solution involves 2 integrations constants, one arbitrary function of x^+ and another of x^-. When the boundary metric is flat and phi=0, it takes a very simple form with the terms at order zero in r charcterized by the arbitrary functions and the subleading term of order r^{-2} by their product. In particular for instance, the BTZ BH corresponds to the simplest solution when these functions reduce to constants. For arbitrary conformally flat boundary metric, the solution is slightly more complicated but not very much so. It has the same structure and involves, besides the same two integration functions, some derivatives of the conformal factor.

6 AdS3/CFT2 Conformal properties asymptotic symmetries transform solutions into solutions g AB = g AB (x, Ξ,ϕ) g AB (x, δξ, δϕ) =L ξ g AB conformal transformation properties δ Y +,Y,ωΞ ±± = Y ± ± Ξ ±± +2 ± Y ± Ξ ±± ±Y ± δ Y +,Y,ωϕ = ω The extended transformation that we have considered before transform solutions to solutions. They thus act on solution space. Explicitly one finds that the arbitrary functions transform exactly like the components of an energy-momentum tensor of a 2 dimensional conformal field theory, besides the conformal weight which is 2,0 and 0,2 respectively, there is an inhomogeneous term involving the third derivative of the transformation parameter.

7 AdS3/CFT2 Charge algebra Hamiltonian approach Q ξ surface charge generators, Dirac algebra centrally extended charge representation of conformal algebra covariant version Q ξ [g ḡ, ḡ] = 1 8πG 2π 0 dφ (Y + Ξ ++ + Y Ξ ) Q ξ1 [L ξ2 g, ḡ] Q [ξ1,ξ 2 ] M [g ḡ, ḡ]+k ξ1,ξ 2, K ξ1,ξ 2 = Q ξ1 [L ξ2 ḡ, ḡ] = 1 8πG 2π 0 dφ ( φ Y τ 1 2 φy φ 2 φy τ 2 2 φy φ 1 ) modes Strominger: combine with Cardy formula to argue for a microscopic derivation of the Bekenstein-Hawking entropy of BTZ black hole What has also been done in the Hamiltonian approach to the problem by Brown & Henneaux is the construction of the generators of the conformal transformations in terms of surface charges, which pair off the integrations constants in the asymptotic symmetries and the solutions and confirms that the Xi s are indeed the components of the energy momentum tensor. Furthermore, they have computed the Dirac brackets of these surface charges and found that they form a centrally extended representation of the conformal algebra. There is a covariant version of this computation which can be used to good effect at null infinity, where a direct Hamiltonian approach is more difficult. What is extremely interesting about this computation is that in a conformal field theory, the value of the central charge contains information about the number of states in the theory. This has been used by Strominger to argue for a microscopic derivation of Bekenstein-Hawking entropy of the BTZ black hole in particular.

8 BMS4/CFT2 Asymptotically flat spacetimes BMS ansatz g µν = Minkowski u = t r η µν = e2β V r + g CDU C U D e 2β g BC U C e 2β 0 0 g AC U C 0 g AB u r r 2 sin 2 θ g AB dx A dx B = r 2 γ AB dx A dx B + O(r) r x A = θ, φ ζ, ζ Sachs: unit sphere γ AB = e 2ϕ 0γ AB 0 γ AB dx A dx B = dθ 2 + sin 2 θdφ 2 Riemann sphere ζ = e iφ cot θ 2, γ ABdx A dx B = e 2 eϕ dζd ζ dθ 2 +sin 2 θdφ 2 = P 2 dζd ζ, P(ζ, ζ) = 1 2 (1 + ζ ζ), ϕ = ϕ ln P determinant condition fall-off conditions det g AB = r4 4 e4 eϕ β = O(r 2 ), U A = O(r 2 ), V/r = 1 2 R + O(r 1 ) Let me now turn to the main topic, which are 4d spacetimes that are asymptotically flat at null infinity. The coordinate choice by Bondi, Metzner and Sachs in the early sixties involves some null coordinate u so that when r goes to infinity at u and the angle fixed, one goes along a light ray. The gauge fixed form of the metric involves zeros for all components involving r except for the g_{ru} component. Minkowski space corresponds to beta and U equal to zero with V=-r and the angle part the metric of the unit sphere. Sachs chooses the leading order of the angular part of the metric to be the one on the unit sphere, but the subsequent geometrical analysis by Penrose in terms of a conformal completion suggests that the analysis can also be performed when keeping an arbitrary conformal factor. One then might as well go to the Riemann sphere by using the standard stereographic coordinates and introduce the conformal factor with respect to the flat metric. Let me stress that none of the conclusions that will be reached depend on this choice of this factor and one could as well keep the original one by Sachs. In this definition of asymptotically flat spacetimes one has only imposed 3 zeros so far and there is then an additional determinant condition on the full angular part which fixes the r coordinate ( luminosity distance ). In addition, there are suitable fall off-conditions on the functions beta, U^A and V.

9 BMS4/CFT2 Asymptotic symmetries asymptotic symmetries general solution L ξ g rr =0, L ξ g ra =0, L ξ g AB g AB =0, L ξ g ur = O(r 2 ), L ξ g ua = O(1), L ξ g AB = O(r), L ξ g uu = O(r 1 ) ξ u = f, f = f ϕ + 1 ξ A = Y A + I A, I A = f,b dr (e 2β 2 ψ f = eϕ T + 1 g AB 2 ), r ξ r = 1 2 r( D A ξ A f,b U B +2f u ϕ), ψ = D A Y A u 0 du e ϕ ψ, Y A = Y A (x B ) T = T (x B ) conformal Killing vectors of the sphere generators for supertranslations spacetime vectors with modified bracket form faithful representation of bms 4 algebra [(Y 1,T 1 ), (Y 2,T 2 )] = ( Y, T ) Y A = Y1 B B Y2 A Y1 B B Y2 A Sachs 1962, T = Y1 A A T 2 Y2 A A T (T 1 A Y2 A T 2 A Y1 A ) standard GR choice: restrict to globally well-defined transformations SL(2, C)/Z 2 SO(3, 1) Y A generators of Lorentz algebra Again asymptotic symmetries are generated by vectors that leave this form of the metric, including the determinant condition, invariant. The general solution can be worked out explicitly and is parametrized by conformal Killing vectors of the unit or equivalently of the Riemann sphere and by an additional arbitrary function T which are the generators of so-called supertranslations on which the conformal Killing vectors act in a suitable way. In the modified bracket, these vectors form a representation of the so-called bms4 algebra, the semi-direct sum of the algebra of conformal Killing vectors of the unit or Riemann sphere acting in a suitable way on the abelian ideal of supertranslations. This result was basically obtained by Sachs as early as 1962 without using this modified bracket. The question is then what are the conformal Killing vectors of the Riemann sphere or the unit sphere. What has been done traditionally in GR is to concentrate on those transformations that are globally well-defined. This singles out SL(2,C) mod Z_2 which is isomorphic to the Lorentz group. In this case, the bms algebra is the semi-direct sum of the Lorentz algebra with the infinite-dimensional abelian ideal of supertranslations.

10 BMS4/CFT2 New proposal CFT choice : allow for meromorphic functions on the Riemann sphere solution to conformal Killing equation Y ζ = Y ζ (ζ), Y ζ = Y ζ( ζ) generators l n = ζ n+1 ζ, ln = ζ n+1 ζ, n Z T m,n = ζ m ζn, m, n Z commutation relations [l m,l n ]=(m n)l m+n, [ l m, l n ]=(m n) l m+n, [l m, l n ]=0, [l l,t m,n ]=( l +1 2 m)t m+l,n, [ l l,t m,n ]=( l +1 2 n)t m,n+l. Poincaré subalgebra l 1,l 0,l 1, l 1, l 0, l 1, T 0,0,T 1,0,T 0,1,T 1,1, But in the context of quantum conformal field theories in 2 dimensions, people have faced the same question and chosen to allow for meromorphic functions on the Riemann sphere. The solution to the conformal Killing equation then involves one arbitrary function of zeta and one of zeta bar. When expanding the supertranslations in the same way in terms of a double Laurent series, the generators of the complexified Lie algebra form the following semi-direct extension of the Virasoro algebra. This algebra is to the Poincare algebra what the Virasoro algebra is to the Lorentz algebra. The Poincaré algebra is of course the subalgebra of exact Killing vectors of Minkowski space and corresponds to the subspace spanned by the generators l_{-1},l_0,l_{1} and their complex conjugates together with T_{0,0}, T_{1,0}, T_{0,1} and T_{1,1}. Before exploring the consequences of this new proposal, let me turn first to the 3d flat case, for which solution space and the charge algebra are much easier to control.

11 BMS3/CFT1 ansatz for asymptotically flat metrics g µν = Asymptotic symmetries e 2β Vr 1 + r 2 e 2ϕ U 2 e 2β r 2 e 2ϕ U e 2β 0 0 r 2 e 2ϕ U 0 r 2 e 2ϕ Minkowski spacetime ds 2 = du 2 2dudr + r 2 dφ 2 u = t r fall-off conditions β = O(r 1 ), U = O(r 2 ) V = 2r 2 u ϕ + O(r) asymptotic symmetries L ξ g rr =0=L ξ g rφ, L ξ g φφ =0, L ξ g ur = O(r 1 ), L ξ g uφ = O(1), L ξ g uu = O(1) ξ u = f, ξ φ = Y + I, I = e 2ϕ φ f dr r 2 e 2β = 1 r r e 2ϕ φ f + O(r 2 ), ξ r = r φ ξ φ φ fu + ξ φ φ ϕ + f u ϕ, u f = f u ϕ + Y φ ϕ + φ Y f = e ϕ T + u 0 du e ϕ ( φ Y + Y φ ϕ) solution involves 2 arbitrary functions on the circle spacetime vector form faithful representation of Y = Y (φ), T = T (φ) bms 3 algebra [(Y 1,T 1 ), (Y 2,T 2 )] = ( Y, T ) Y = Y 1 φ Y 2 (1 2), T = Y1 φ T 2 + T 1 φ Y 2 (1 2) In the flat 3d case, the analog of the FG ansatz for asymptotically flat metrics at null infinity is the same as for the 4d case, but with one less angle. As a consequence of the determinant condition, the angle part of the metric is completely fixed in terms of a conformal factor that depends only on u and the angle but not on r. Again, the asymptotic symmetries are generated by the vector fields that leave this form of the metric invariant. These equations can be exactly solved. In the case of an u dependent conformal factor, the solution is sligthly involved, but the precise form is not so important. What matters is that it contains two arbitrary functions on the circle Y and T with all subleading terms determined in terms of the metric. In the modified bracket, these vectors form a faithful representation of the so-called bms3 algebra. It consists of the semi-direct sum of the vector fields on the circle equipped with the usual commutator acting on functions on the circle.

12 BMS3/CFT1 Solution space and conformal properties general solution parametrized by Θ = Θ(φ), Ξ = Ξ(φ) s uφ = e ϕ Ξ + u 0 ds 2 = s uu du 2 2dudr +2s uφ dudφ + r 2 e 2ϕ dφ 2, s uu = e 2ϕ Θ ( φ ϕ) φϕ 2r u ϕ, du e ϕ 1 2 φθ φ ϕ[θ ( φ ϕ) φϕ]+ 3 φϕ. bms 3 transformation properties δ Y,T Θ = Y φ Θ +2 φ Y Θ 2 3 φy, δ Y,T Ξ = Y φ Ξ +2 φ Y Ξ T φθ + φ T Θ 3 φt, covariant charges Q ξ [g ḡ, ḡ] 1 16πG K ξ1,ξ 2 = 1 8πG 2π 0 dφ 2π 0 dφ (ΘT +2ΞY ) φ Y 1 (T 2 + φt 2 2 ) φ Y 2 (T 1 + φt 2 1 ) It turns out that the general solution to the equations of motion can also be derived. It is even easier than in the AdS3 case and involves, besides the conformal factor, 2 arbitray functions on the circle. The transformation properties can be computed and involve inhomogeneous terms, but not in the standard way. When computing the charges, one finds that they pair off the arbitrary functions in the solution with those in the asymptotic symmetries. Their covariant Poisson algebra is again centrally extended.

13 BMS3/CFT1 Charge algebra modes Y (θ) 1 copy of Wit algebra acting on i 1 iso(2, 1) charge algebra: relation to AdS 3 similar to contraction between so(2, 2) iso(2, 1) L ± m = 1 2 ( lp ±m ± J ±m ) l collaboration with G. Compère After Fourier analyzing, the algebra consists of 1 copy of the Wit algebra acting on the functions on the circle in a similar way than the Lorentz transformations act on the ordinary translations. Poincaré transformations in 3d are obtained by restricting to the subalgebra of generators with labels -1,0,1. In fact the bms3 algebra has been originally discussed in a paper by Ashtekar et al in When one now expresses the central extension in terms of these generators, it appears not in the Virasoro factor, but between the two factors describing the Witt algebra and the supertranslations. A posteriori, it is clear that this is the only place for the central extension as it cannot appear in the one copy of the Wit algebra on account of the missing dimensional parameter l in the flat case. The relation to the AdS3 Virasoro case is by a contraction similar to the relation between so(2,2) and iso(2,1). More precisely, if one introduces a parameter of dimension length, there is an extension of the BMS algebra, and after redefining the generators, one finds both for the asymptotic symmetries and for the charges, including the central ones, the AdS3 results. What would be interesting is to analyze in details what this classical central extension can teach us about quantum gravity in asymptotically flat 3d spacetimes. One should study whether a Cardy type formula works in this case also. Notice also that there are no black holes, but only conical defect solutions in this case.

14 BMS4/CFT2 solution space ansatz g AB = r 2 γ AB + rc AB + D AB γ ABC C DC D C + o(r ) determinant condition C A A =0=D A A Sachs: power series and D AB =0 guarantees absence of log terms equations of motion imply β = β(g AB ) U A = 1 2 r 2 DB C BA 2 3 r 3 (ln r ) D B D BA 1 2 CA B D C C CB + N A + o(r 3 ε ), angular momentum aspect N A (u, x A ) u dependence fixed log terms also absent when D ζζ = d(ζ), D ζ ζ = d( ζ), D ζ ζ =0. In the 4d case, solution space depends crucially on what assumptions one imposes on the angular part of the metric. Let us use the following set-up, where there are no log terms up to terms that go to zero faster than any small power of r. Because of the determinant condition both C and D are traceless. In the original work, Sachs assumed a powers series dependence and the absence of the D terms. This guarantees the absence of log terms in the whole solution. The equations of motion then fix completely beta in terms of the angular metric. In the determination of U^A there arise integrations functions N^A, the angular momentum aspect, which depend really on the angles because the u dependence is fixed by a suitable differential equation. In the zeta,bar zeta parametrization, one then sees that the leading log terms are also absent if the two components of D are respectively holomorphic and anti-holomorphic functions.

15 BMS4/CFT2 Solution space V r = 1 2 R + r 1 2M + o(r 1 ) mass aspect M(u, x A ) u dependence fixed news tensor u C AB (u, x A ) only arbitray function of u general solution: 4 arbitrary functions of 3 variables & 3 arbitrary functions of 2 variables g AB (u 0,r,x A ) u C AB (u, x A ) M(u 0,x A ) N A (u 0,x A ) for simplicity ϕ =0 γ AB dx A dx B = dζd ζ C ζζ = c, C ζ ζ = c, C ζ ζ =0 redefinitions M = M 2 c 2 c Ñ ζ = 1 12 [2N ζ +7 c c +3c c] evolution equations u M = ċ c 3 u Ñ ζ = M 2 3 c ( c +3 c )ċ The mass aspect then arises as an integration function in the determination of the function V, and again, its u dependence is fixed by a suitable differential equation. Finally there is an equation determining the r dependence of the u derivative of g_{ab}. The integration constant for this equation is news tensor which agrees here with the time derivative of C_A{B}. It is the only arbitrary function of u that is left in the problem. The general solution of the equations of motion then contains 4 arbitray functions of 3 variables and 3 functions of 2 variables. For simplicity, let us for instance consider the conformal factor with respect to the metric on the Riemann sphere to vanish and let us denote the non vanishing components of the C tensor by c and \bar c respectively. After some redefinitions involving these components, the evolution equations fixing the tme dependence of the modified mass aspect only depends on the news tensor while the one for the angular momentum aspects is more complicated.

16 BMS4/CFT2 Conformal properties bms4 transformations δc = fċ + Y A A c +( 3 2 Y 1 2 Ȳ )c 2 2 f δd = Y A A d +2 Y d f = T uψ δċ = f c + Y A A ċ +2 Y ċ 3 Y δ M = fċ c + Y A A M ψ M + c 3 Y + c 3 Ȳ T δñ ζ = Y A A Ñ ζ +( Y +2 Ȳ )Ñ ζ + 1 (ψ d) 3 f( M +2 2 c + cċ) f M c +( c +3 c )ċ The properties of the integration constants under bms4 transformations, and thus also in particular under local conformal transformations, can then be worked out. We can read off conformal weights and inhomogeneous terms. For instance, in the case where d is holomorphic, we see that it is a primary field of weight 2, whereas the transformation properties of the news tensor look very much like those of an energy momentum tensor. What is also interesting is the that the Schwartzian derivative terms in the transformation of the modified mass aspect involve the fields c and \bar c instead of constants.

17 In any case, the new bms4 algebra seems really rather appealing and appears as a perfectly natural infinitedimensional extension of the Poincaré algebra... BMS4/CFT2 Conclusions and perspectives 4d gravity is dual to some conformal field theory classifiy (non)-central extensions; study representation theory of bms4 to be done: surface charge algebra non extremal Kerr/CFT correspondence? angular momentum problem in GR: Lorentz = bms4(old)/supertranslations versus bms4(new)/supertranslations = Virasoro As a conclusion, we can interpret the obtained results in the sense that 4d gravity is dual to some kind of conformal field theory. What remains to be done is to better understand this theory. Let me recall that even in the AdS3 case, this point does not yet seem to be completely understood. In order to get a better understanding, one should first work out the representation theory for the new version of bms4, and it should be much better behaved than the one for the old version of bms4. Indeed, algebra has a similar semi-direct structure than the Poincaré algebra, and the representation theoretic techniques are well-developed, while the representation theory for the Virasoro algebra is also well-known. One should also classify possible central and non-central extensions by computing the associated Lie algebra cohomology groups. Another work in progress concerns the algebra of surface charges associated with the asymptotic symmetries. One aim would be to use the results on the transformation properties of the various fields to understand something about a non-extremal Kerr/CFT correspondence for instance. The new proposal for bms4 also sheds a new light on the problem of angular momentum in GR. Indeed, for the old version, the Lorentz algebra appears as the quotient of bms4 by the abelian ideal of supertranslations. It follows that to define out angular momentum one needs to fix a point, a representative for the translations, which requires 4 conditions in the standard Poincaré case, but an infinite number of them when there are supertranslations. The problem is completely different for the new bms4 algebra as in this case the quotient space is the infinite dimensional Virasoro algebra and could very well go away completely. But all this is very much work in progress.

18 References Asymptotically flat spacetimes & symmetries

19 References Gravitational AdS3/CFT2 & Kerr/CFT

20 References

21 References Holography at null infinity in 3 & 4 dimensions

22 References This work based on