A d-dimensional stress tensor for Minkd+2 gravity

Journal of High Energy Physics, May 2018

Abstract We consider the tree-level scattering of massless particles in (d+2)-dimensional asymptotically flat spacetimes. The \( \mathcal{S} \)-matrix elements are recast as correlation functions of local operators living on a space-like cut ℳ d of the null momentum cone. The Lorentz group SO(d + 1, 1) is nonlinearly realized as the Euclidean conformal group on ℳ d . Operators of non-trivial spin arise from massless particles transforming in non-trivial representations of the little group SO(d), and distinguished operators arise from the soft-insertions of gauge bosons and gravitons. The leading soft-photon operator is the shadow transform of a conserved spin-one primary operator J a , and the subleading soft-graviton operator is the shadow transform of a conserved spin-two symmetric traceless primary operator T ab . The universal form of the soft-limits ensures that J a and T ab obey the Ward identities expected of a conserved current and energy momentum tensor in a Euclidean CFT d , respectively.

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A d-dimensional stress tensor for Minkd+2 gravity

Accepted: May Minkd+2 gravity Daniel Kapec 0 2 3 Prahar Mitra 0 1 2 3 Quantum Gravity, Scattering Amplitudes 0 Center for the Fundamental Laws of Nature, Harvard University 1 School of Natural Sciences, Institute for Advanced Study 2 Cambridge , MA, 02138 U.S.A 3 Princeton, NJ , 08540 U.S.A We consider the tree-level scattering of massless particles in (d+2)-dimensional asymptotically at spacetimes. The S-matrix elements are recast as correlation functions of local operators living on a space-like cut Md of the null momentum cone. The Lorentz group SO(d + 1; 1) is nonlinearly realized as the Euclidean conformal group on Md. Operators of non-trivial spin arise from massless particles transforming in non-trivial representations of the little group SO(d), and distinguished operators arise from the soft-insertions of gauge bosons and gravitons. The leading soft-photon operator is the shadow transform of a conserved spin-one primary operator Ja, and the subleading soft-graviton operator is the shadow transform of a conserved spin-two symmetric traceless primary operator Tab. The universal form of the soft-limits ensures that Ja and Tab obey the Ward identities expected of a conserved current and energy momentum tensor in a Euclidean CFTd, respectively. Conformal Field Theory; Field Theories in Higher Dimensions; Models of - A 1 Introduction 2 3 4 Lorentz transformations and the conformal group Conserved currents and soft theorems 4.1 4.2 4.3 Leading soft-photon theorem and the conserved U(1) current Subleading soft-graviton theorem and the stress tensor Leading soft-graviton theorem and momentum conservation 5 Conclusion A Spacetime picture 1 Introduction [ I , is a null cone with a (possibly singular) vertex at spatial in nity. Massless excitations propagating in such a spacetime pass through I at isolated points on the celestial sphere. Guided by the holographic principle, one might hope that the S-matrix for the scattering of massless particles in asymptotically at spacetimes in (d + 2)-dimensions might be reexpressed as a collection of correlation functions of local operators on the celestial sphere Sd at null in nity, with operator insertions at the points where the particles enter or exit the spacetime. The Lorentz group would then be realized as the group of conformal motions of the celestial sphere, and the Lorentz covariance of the S-matrix would guarantee that the local operators have well-de ned transformation laws under the action of the Euclidean conformal group SO(d + 1; 1). On these general grounds one expects the massless S-matrix to display some of the features of a d-dimensional Euclidean conformal eld theory (CFTd). + It has recently become possible to make some of these statements more precise in four dimensions, due in large part to Strominger's infrared triangle that relates soft theorems, asymptotic symmetry groups and memory e ects [1{40]. While the speci c details of a putative holographic formulation are expected to be model dependent, it should be possible to make robust statements (primarily regarding symmetries) based on universal properties of the S-matrix. One interesting class of universal statements about the S-matrix concerns the so-called soft-limits [41{48] of scattering amplitudes. In the limit when the wavelength of an external gauge boson or graviton becomes much larger than any scale in the scattering process, the S-matrix factorizes into a universal soft operator (controlled by the soft particle { 1 { and the quantum numbers of the hard particles) acting on the amplitude without the soft insertion. This sort of factorization is reminiscent of a Ward identity, and indeed in four dimensions the soft-photon, soft-gluon, and soft-graviton theorems have been recast in the form of Ward identities for conserved operators in a putative CFT2 [2, 16, 28, 31, 34, 49, 50]. Most importantly for the present work, in [28, 31, 50] an operator was constructed from the subleading soft-graviton theorem whose insertion into the four dimensional Smatrix reproduces the Virasoro Ward identities of a CFT2 energy momentum tensor. The subleading soft-graviton theorem holds in all dimensions [51{57], so it should be possible to construct an analogous operator in any dimension. We will see that this is indeed the case, and that the construction is essentially xed by Lorentz (conformal) invariance. massless particle kinematics and describe the map from the (d + 2)-dimensional S-matrix to a set of d-dimensional \celestial correlators" de ned on a space-like cut of the null momentum cone. Section 3 describes the realization of the Euclidean conformal group on these correlation functions in terms of the embedding space formalism. Section 4 outlines the construction of conserved currents | namely the conserved U(1) current and the stress tensor | in the boundary theory and their relations to the leading soft-photon and subleading soft-graviton theorems. Section 5 concludes with a series of open questions. In appendix A, we brie y discuss the bulk space-time interpretation of our results and their relations to previous work. 2 Massless particle kinematics The basic observable in asymptotically at quantum gravity is the S-matrix element S = h out j in i (2.1) (2.2) (2.3) between an incoming state on past null in nity ( I ) and an outgoing state on future null in nity (I+). The perturbative scattering states in asymptotically at spacetimes are characterized by collections of well separated, non-interacting particles.1 Each massless particle is characterized by a null momentum p and a representation of the little group SO(d), as well as a collection of other quantum numbers such as charge, avor, etc. Null space Rd+1;1, momenta are constrained to lie on the future light cone C + of the origin in momentum C + = fp 2 Rd+1;1 p2 = 0 ; p0 > 0g : A convenient parametrization for the momentum, familiar from the embedding space formalism in conformal eld theories [58, 59], is given by p (!; x) = ! (x)p^ (x) ; p^ (x) = 1In four dimensions, the probability to scatter into a state with a nite number of gauge bosons or where x2 = xaxa = abxaxb. The metric on this null cone is degenerate and is given by (x) de nes the conformal factor on Md. In most of what follows we will choose (x) = 1 for computational simplicity, although the generalization to an arbitrary conformally at Euclidean cut is straightforward.2 The (d+2)-dimensional Lorentz-invariant measure takes while the Lorentzian inner product is given by Massless particles of spin s can be described by symmetric traceless elds "a11::::::ass (p) Oa1:::as (p)eip X + Oay1:::as (p)e ip X i h satisfying the equations Under gauge transformations, We will work in the gauge X 1::: s (X) = 0 ; n 1 1::: s (X) = 0 ; n = (1; 0a; 1) : A natural basis for the vector representation of the little group SO(d) is given in terms of the d polarization vectors "a (x) These are orthogonal to both n and p^ and satisfy "a(x) "b(x) = ab ; "a (x)"a(x) = (x) + n p^ (x) + n p^ (x) : 2Other choices of (x) have also proved useful in previous analyses. In particular, the authors of [3, 4, 8, 18] choose (x) = 2(1 + x2) 1, yielding the round metric on Sd. For non-constant (x), d-dimensional by their conformally covariant counterparts, the GJMS operators [60]. 1 h"a (x)"b (x) + "b (x)"a(x) i 1 ab d (x) : The Fock space of massless scattering states is generated by the algebra of single the energy direction actually appears space-like rather than time-like.3 The creation and annihilation operators can be viewed as operators inserted at speci c points of Md carrying an additional quantum number !, so that the S-matrix takes the form of a conformal correlator of primary operators. In other words, the amplitude with m incoming and n m outgoing particles4 An;m = hpm+1; : : : ; pnjp1; : : : ; pmi can be equivalently represented as a correlation function on Md An = h O1(p1) : : : On(pn) iMd = h O1(!1; x1) : : : On(!n; xn) iMd : In this representation, outgoing states have ! > 0 and ingoing states have ! < 0. In the rest of this paper, we will freely interchange the notation pi , (!i; xi) to describe the insertion of local operators. In appendix A, we demonstrate that the space-like cut Md of the momentum cone is naturally identi ed with the cross sectional cuts of I+. We construct the bulk coordinates whose limiting metric on future null in nity is that of a null cone with cross-sectional metric (2.5). This provides a holographic interpretation of our construction, recasting scattering amplitudes in asymptotically at spacetimes in terms of a \boundary" theory that lives on I+. 3This is also the case for the null direction on I in asymptotic quantization. 4Here, we have suppressed all other quantum numbers that label the one-particle states, such as the polarization vectors, avor indices, or charge quantum numbers. { 4 { (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) In this section, we make explicit the map from (d + 2)-dimensional momentum space Smatrix elements to conformal correlators on the Euclidean manifold Md. The setup is mathematically similar to the embedding space formalism, although in this case the (d+2)dimensional \embedding space" is the physical momentum space rather than merely an auxiliary ambient construction. The Lorentz group SO(d + 1; 1) with generators M acts linearly on momentum space vectors p group of Md is given by the identi cations 2 Rd+1;1. The isomorphism with the conformal Jab = Mab ; Ta = M0;a Md+1;a ; D = Md+1;0 ; Ka = M0;a + Md+1;a : (3.1) The Jab generate SO(d) rotations, D is the dilation operator, and Ta and Ka are the generators of translations and special conformal transformations, respectively. These operators satisfy the familiar conformal algebra [Jab; Jcd] = i( acJbd + bdJac bcJad adJbc) ; HJEP05(218)6 Equation (3.1) describes the precise map between the linear action of Lorentz transformations on p and nonlinear conformal transformations on (!; x). The latter amount to transformations x ! x0(x) for which Using (3.1), the conformal properties of the operators O(!; x) (we suppress spin labels for convenience) follow from their Lorentz transformations, [O(p); M ] = L O(p) + S O(p) ; [O(p); P ] = p O(p) ; denotes the spin-s representation of the Lorentz group. We would like to rewrite these relations in a way that manifests the action on Md. First, we note that 2xa ; 2xaxb 2 along with where and S [Jab; Tc] = i( acTb [Jab; Kc] = i( acKb [D; Ta] = iTa ; [D; Ka] = iKa ; bcTa) ; bcKa) ; [Ta; Kb] = 2i( abD + Jab) : ! ! !0 = ! (x) : L = i p (3.5) (3.6) 2xa La;d+1 = i 2 i 2 (1 Lab = (3.9) HJEP05(218)6 (3.8) (3.10) (3.11) (3.12) The action of the spin matrix S can be conveniently expressed in terms of the polarization vectors (2.12). For instance, the action on a spin-1 state is given by In general one nds [ where Sab is the representation of the massless little group SO(d). The action of the conformal group on the creation and annihilation operators is then given by [O(!; x); Jab] = We recognize these commutation relations as the de ning properties of a spin-s conformal primary operator, with a non-standard dilation eigenvalue = the momentum cone along its null direction.5 The fact that is realized as a derivative simply re ects the fact that the energy eigenstates do not diagonalize the dilation operator, which simply translates the space-like cut Md of 4 Conserved currents and soft theorems The operator content and correlation functions of the theory living on Md are highly dependent on the spectrum and interactions of the (d + 2)-dimensional theory under consideration. However, the universal properties of the (d + 2)-dimensional S-matrix are expected to translate into general, model independent features of the \boundary theory". In this section, we explore the consequences of the universal soft factorization properties of Smatrix elements. Soft factorization formulas closely resemble Ward identities, and indeed many soft theorems are known to be intimately related to symmetries of the S-matrix. The existence of the soft theorems should therefore enable one to construct associated conserved currents for the theory living on Md. The appropriate currents were constructed for d = 2 in [2, 16, 28, 31, 34, 50]. Here, we generalize these results to d > 2. RC d!! 1O(!; x) for some contour C in the complex ! plane. 5It is possible to obtain standard conformal primary operators via a Mellin transform, O( ; x) = { 6 { The leading soft-photon theorem is a universal statement about the behavior of S-matrix elements in the limit that an external photon's momentum tends to zero. It is model independent, exists in any dimension, and states that h Oa(q)O1(p1) : : : On(pn) i n From (4.3), we see that Sa(x) is a conformal primary operator with ( ; s) = (1; 1). Note also that Sa(x) satis es identically, without contact terms. In even dimensions,6 the leading soft-photon theorem is known [8, 12, 13, 17, 18] to be completely equivalent to the invariance of the S-matrix under a group of angle dependent U(1) gauge transformations with non-compact support. In d = 2, this symmetry is generated by the action of a holomorphic boundary current Jz satisfying the appropriate Kac-Moody Ward identities (see [16]). In higher dimensions, one consequently expects to encounter a conformal primary operator Ja(x) with ( ; s) = (d 1; 1) satisfying the Ward identity h @bJb(y)O1(!1; x1) : : : On(!n; xn) i = X Qk (d)(y xk)h O1(!1; x1) : : : On(!n; xn) i : n k=1 n k=1 { 7 { 6From this point on, we will consider only the even dimensional case in order to avoid discussion of fractional powers of the Laplacian. operator" the form where Oa(!; x) creates an outgoing photon of momentum p(!; x) and polarization "a (x), and Qk is the charge of the k-th particle. We will rst de ne the \leading soft-photon HJEP05(218)6 Sa(x) = !li!m0 !Oa(!; x) : Insertions of this operator are controlled by the leading soft-photon theorem (4.1) and take h Sa(x)O1(!1; x1) : : : On(!n; xn) i = @a X Qk log (x xk)2 h O1(!1; x1) : : : On(!n; xn) i: Our goal is to construct the conserved current Ja(x) from the soft operator Sa(x). The inverse problem | constructing Sa(x) from an operator Ja(x) satisfying (4.5) | is easily y)2], we nd Z xk)2]h O1(!1; x1) : : : On(!n; xn) i = h Sa(x)O1(!1; x1) : : : On(!n; xn) i : (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) We therefore identify7 where Iab(x y) is the conformally covariant tensor Z Sa(x) = Z ddy Iab(x (x y)y2) J b(y) ; Iab(x y) = ab 2 (x y)a(x (x y)2 y)b : This nonlocal relationship between the = 1 primary Sa and the = d 1 primary Ja is known as a shadow transform. For a spin-s operator of dimension , the shadow operator HJEP05(218)6 is given by [58] Oea1:::as (x) = ab11::::::bass Z ddy Ib1c1 (x [(x y)2]d y) : : : Ibscs (x y) Oc1:::cs (y) : Here, ab11::::::bass is the invariant identity tensor in the spin-s representation, ab11::::::bass = ffab11 ab22 : : : abssgg ; where the notation f ; g denotes the symmetric traceless projection on the indicated indices. The shadow transform is the unique integral transform that maps conformal primary operators with ( ; s) onto conformal primary operators with (d ; s). Given that Sa has ( ; s) = (1; 1) while Ja has ( ; s) = (d 1; 1) it seems natural to expect the appearance of the shadow transform. The shadow transform is, up to normalization [58, 61], its own inverse8 (4.7) (4.8) (4.9) (4.10) d Using this, we can immediately write ( d ( 1)(d 1 + s)(d 1) d 2 1 + s) ( ) (d Sa(x) = 2Jea(x) ; 1 Ja(x) = 2c(1; 1) Sea(x) : Interestingly, the property (4.4) allows one to obtain a local relation between Ja(x) and Sa(x) Ja(x) = 1 (4 )d=2 (d=2) ( ) d2 1Sa(x) : It is straightforward to verify that insertions of Ja(x) are given by hJa(x)O1(!1; x1) : : : On(!n; xn)i = 2 d=2 (d=2) Xn Qk x (x k=1 j xk)a xkjd h O1(!1; x1) : : : On(!n; xn) i and satisfy (4.5). which do not a ect the Ward identity (4.5). 8The spatial integrals involved here are formally divergent and are regulated by the i -prescription. { 8 { In summary, we nd that the leading soft-photon theorem in any dimension implies the existence of a conserved current Ja(x) on the spatial cut Md. This current is constructed as the shadow transform of the soft-photon operator Sa(x). This correspondence is reminiscent of a similar construction in AdS/CFT, where again the presence of a massless bulk gauge eld produces a dual conserved boundary current. There have been attempts to make this analogy more precise using a so-called \holographic reduction" of Minkowski space [50, 62]. The (d + 2) dimensional Minkowski space can be foliated by hyperboloids (and a null cone), each of which is invariant under the action of the Lorentz group. Inside the light cone, this amounts to a foliation using a family of Euclidean AdSd+1, all sharing an asymptotic boundary given by the celestial sphere.9 Performing a Kaluza-Klein reduction on the (time-like, non-compact) direction transverse to the AdSd+1 slices decomposes the gauge eld A(X) in Minkowski space into a continuum of AdSd+1 gauge elds A!(x) with masses ! (! is the so-called Milne energy). The ! ! 0 gauge eld | equivalent to the soft limit | is massless in AdSd+1 and therefore induces a conserved current on the d-dimensional boundary. The holographic dictionary suggests that in the boundary theory, one has a coupling of the form Z ddxSa(x)Ja(x) : (4.15) The discussion here suggests that a deeper holographic connection (beyond the simple existence of conserved currents) may exist between the theory on Md and dynamics in Minkowski space. While intriguing, much remains to be done in order to elucidate this relationship. The hypothetical boundary theory is expected to have many peculiar properties, one of which we discuss in section 4.3. 4.2 Subleading soft-graviton theorem and the stress tensor In the previous section, we demonstrated that the presence of gauge elds in Minkowski space controls the global symmetry structure of the putative theory on Md. As in AdS/ CFT, more interesting features arise when we couple the bulk theory to gravity and consider gravitational perturbations. Flat space graviton scattering amplitudes also display universal behavior in the infrared that is model independent and holds in any dimension. Of particular interest here is the subleading soft-graviton theorem, which states10 k=1 i Xn "ab pk q pk q !!0 Jk hO1(p1) : : : On(pn)i : (4.16) Here Oab(q) creates a graviton with momentum q and polarization "ab (q), and Jk is the total angular momentum operator for the k-th particle. The operator (1 + !@!) projects out the Weinberg pole [45], yielding a nite ! ! 0 limit. 9This construction seems more natural in momentum space, where it is only the interior of the light cone which is physically relevant, and one never needs to discuss the time-like de Sitter hyperboloids lying outside the light cone. 10We work in units such that p8 G = 1. { 9 { Returning to the analogy with AdSd+1=CFTd, one might expect that the bulk softgraviton is associated to a boundary stress tensor, just as the bulk soft-photon is related to a boundary U(1) current. In a quantum eld theory, the stress tensor generates the action of spacetime (conformal) isometries on local operators. As we saw in (3.11), the angular momentum operator J generates these transformations on the local operators on Md. Therefore, it is natural to suspect that the bulk subleading soft-graviton operator !!0 is related to the boundary stress tensor. Such a relationship was derived in four dimensions Insertions of Bab(x) are controlled by the subleading soft-graviton theorem (4.16) and take the form (see (3.8) and (3.10) for the explicit forms of the orbital and spin angular momentum operators) hBab(x)O1(!1; x1) : : : On(!n; xn)i = Pcab(x From (4.18), we see that Bab(x) is a conformal primary operator with ( ; s) = (0; 2). One can also check that As in section 4.1, it is easiest to rst determine Bab in terms of Tab. Recall that the Ward identities for the energy momentum tensor of a CFTd take the form [63] h @dTdc(y)O1(!1; x1) : : : On(!n; xn) i = (d)(y xk)@xck h O1(!1; x1) : : : On(!n; xn) i ; where Pcab(x) = 1 2 xa bc + xb ac + 2 c d x ab x2 4 xcxaxb : h T cc(y)O1(!1; x1) : : : On(!n; xn) i = (d)(y xk)!k@!k h O1(!1; x1) : : : On(!n; xn) i ; h T [cd](y)O1(!1; x1) : : : On(!n; xn) i = (d)(y xk)Skcdh O1(!1; x1) : : : On(!n; xn) i : n X k=1 2 n X k=1 n X k=1 i Xn 2 k=1 1 xk)Skcd hO1(!1; x1) : : : On(!n; xn)i ; (4.17) (4.18) (4.19) (4.20) (4.21) (4.22) (4.23) Multiplying (4.21) by y), and taking the sum, one nds Tefabg(x) : Bab(x) = y)T cd(y) ddyIfafc(x y)Idgbg(x y)T cd(y) Once again, the soft operator appears as the shadow transform of a conserved current. The relationship could have been guessed from the outset based on the dimensions of Bab and Tfabg.11 Having derived (4.24), we can now invert the shadow transform to nd Tfabg(x) = 1 c(0; 2) Beab(x) : Z ddxBab(x)Tab(x) The shadow relationship between the soft operator Bab and the energy momentum tensor is again suggestive of a coupling in some hypothetical dual formulation of asymptotically at gravity: the soft-graviton creates an in nitesimal change in the boundary metric, sourcing the operator Tab. In [28], it was viewed as a puzzle that the energy momentum tensor appears non-local when written in terms of the soft-modes of the four dimensional gravitational eld. Here we see that this is essentially the consequence of a linear response calculation, and that the non-locality is actually the only one allowed by conformal symmetry. As in [28], it is possible to derive a local di erential equation for Tfabg in even dimensions. We rst de ne the following derivative operator DaOab 1 2(4 )d=2 (d=2 + 1) One can check that Then, acting on the rst equation of (4.24) with Da, we nd d d (4.27) DaPcab = c (d)(x) : b 11Note that only the symmetric traceless part of the stress tensor appears in this dictionary since the graviton lies in the symmetric traceless representation of the little group. The trace term may be related to soft-dilaton theorems. (4.24) (4.25) (4.26) (4.28) (4.29) In the previous two subsections, we have avoided the discussion of currents related to the leading soft-graviton theorem. This soft theorem is associated to spacetime translational invariance (and more generally to BMS supertranslations [3, 4, 15, 20]). For scattering amplitudes in the usual plane wave basis, this symmetry is naturally enforced by a momentum conserving Dirac delta function. In our discussion above, we have chosen to make manifest the Lorentz transformation properties of the scattering amplitude. Consequently, translation invariance, or global momentum conservation, is unwieldy in our formalism. In fact, it must somehow appear as a non-local constraint on the correlation functions on Md, since arbitrary operator insertions corresponding to arbitrary con gurations of incoming and outgoing momenta will in general violate momentum conservation. The di culty can also be seen at the level of the symmetry algebra. Momentum conservation cannot arise simply as a global Rd+1;1 symmetry of the CFTd, since the associated conserved charges do not commute with the conformal (Lorentz) group. In light of this it is not clear that our construction can really be viewed as a local conformal eld theory living on Md.12 We have tried, unsuccessfully, to nd a natural set of operators whose shadow reproduces the leading soft-graviton theorem13 !li!m0 !hOab(q)O1(p1) : : : On(pn)i = ! Xn "ab p p k k pk q hO1(p1) : : : On(pn)i : k=1 The soft operator has insertions given by Gab(x) = lim !Oab(!; x) !!0 h @aUa(x)O1(!1; x1) : : : On(!n; xn)i = xk)hO1(!1; x1) : : : On(!n; xn)i : 12However, it is also not clear that we should expect a local QFT dual to asymptotically at quantum gravity. The at space Bekenstein-Hawking entropy is always super-Hagedorn in d 4. The high energy density of states grows faster than in any local theory. 13It has also been suggested [64] that translational invariance of the S-matrix is realized through null state relations of boundary correlators rather than through local current operators, since the former are typically non-local constraints on CFT correlation functions. k=1 d 2 Xn !kIa(db)(x d xaxb x2 : 1) ( (d)(x n X !k k=1 (4.30) (4.31) (4.32) (4.33) (4.34) (4.35) Ia(db)(x) = ab 1 (4 )d=2 (d=2) (d hGab(x)O1(!1; x1) : : : On(!n; xn)i = xk)hO1(!1; x1) : : : On(!n; xn)i ; Thus, Ua(x) satis es the current Ward identity corresponding to \energy" conservation. However, since ! is not a scalar charge, the current Ua(x) is not a primary operator (though it has a well-de ned scaling dimension = d). Acting on (4.35) with 2 R ddxIa(db)(y d x), one nds Gab(x) = (4.36) Unlike the U(1) current and the stress tensor, the leading soft-graviton \current" is not related to the soft operator Gab through a shadow transform. It may be possible to interpret (4.36) as some other (conformally) natural non-local transform of Uc(x), but we do not pursue this here.14 5 In this paper we have taken steps to recast the (d + 2)-dimensional S-matrix as a collection of celestial correlators, but many open questions remain. Our analysis relied on symmetry together with the universal behavior of the S-matrix in certain kinematic regimes. It would be interesting to analyze the consequences of other universal properties of the S-matrix for the celestial correlators. The analytic structure and unitarity of the S-matrix should be encoded in properties of these correlation functions, although the mechanism may be subtle. It seems likely that the collinear factorization of the S-matrix could be used to de ne some variant of the operator product expansion for local operators on the light cone. Although this paper only addressed single soft insertions, double soft limits, appropriately de ned, could be used to de ne OPE's between the conserved currents and stress tensors. We expect supergravity soft theorems to yield a variety of interesting operators, including a supercurrent. Finally, the interplay of momentum conservation with the CFTd structure requires further clari cation. We leave these questions to future work. Acknowledgments We are grateful to Sabrina Pasterski, Abhishek Pathak, Ana-Maria Raclariu, Shu-Heng Shao, Andrew Strominger, Xi Yin, and Sasha Zhiboedov for discussions. This work was supported in part by DOE grant DE-SC0007870 and the Fundamental Laws Initiative at Harvard. PM gratefully acknowledges support from DOE grant DE-SC0009988. A Spacetime picture In this appendix we construct coordinates for Minkd+2 whose limiting metric on I cross sectional cuts given by Md. Consider the coordinate transformation from the at + has 14The shadow transform ( ; s) ! (d ; s) is related to the Z2 symmetry of the quadratic and quartic Casimirs of the conformal group c2 = (d ) + s(2 d s), c4 = s(2 d and c4 are also invariant under another Z2 symmetry under which ( ; s) ! (1 s)( s; 1 1)(d 1). c2 ). Equation (4.36) may be the integral representation of a shadow transform followed by the second Z2 transform which maps (d + 1; 0) ! ( 1; 0) ! (1; 2). to the coordinates (u; r; xa) given by where p (xa) is given by (2.3) with ! set to one. We have X (u; r; xa) = rp (xa) + uk (xa) ; dp dp = 2(x)dxadxa : If the vector k (xa) is chosen to satisfy then one nds dX dX = k2du2 + 2(p k)dudr + r2dp dp : metric on I + with cross-sectional metric Since neither p nor k scales with r, the limit r ! 1 with u xed yields a degenerate For instance, the at metric on Md corresponds to the choice ds2 = r2 2(x) abdxadxb = r2ds2Md : p (x) = which yields the familiar coordinate transformation and a metric of the form X (u; r; xa) = u + r(1 + x2) ; rxa; r(1 x2) 2 u ds2 = dudr + r2 abdxadxb : (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) HJEP05(218)6 In order to achieve cross sectional cuts of I+ which are metrically Sd, one chooses p (x) = 2 The coordinate transformation is then and the spacetime metric is X (u; r; xa) = u + r; 12+rxxa2 ; r 1 In order to make the relationship between Md and I coordinate system (A.7). A massless eld in the plane wave basis takes the form + even more explicit, consider the at "a11::::::ass (q) hOa1:::as (q)eiq X + Oay1:::as (q)e iq X i : (A.12) so that X ! rp (xa). In this limit, the argument of the exponential To perform an asymptotic analysis near I+, one considers the limit r ! 1 with u xed, ir!qq(ya) p(xa) = r!q(x y)2 i 2 is large so that the exponential is rapidly oscillating. At leading order in 1r , the only momenta that contribute to the integral are those for which the phase is stationary, i.e. for which x = y. Therefore in the large-r light-like limit, one e ectively trades the transverse coordinates on I + for the momentum coordinates on Md. Armed with this knowledge we can further elaborate on the results of section 4. 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Daniel Kapec, Prahar Mitra. A d-dimensional stress tensor for Minkd+2 gravity, Journal of High Energy Physics, 2018, 186, DOI: 10.1007/JHEP05(2018)186