4D scattering amplitudes and asymptotic symmetries from 2D CFT

Journal of High Energy Physics, Jan 2017

We reformulate the scattering amplitudes of 4D flat space gauge theory and gravity in the language of a 2D CFT on the celestial sphere. The resulting CFT structure exhibits an OPE constructed from 4D collinear singularities, as well as infinite-dimensional Kac-Moody and Virasoro algebras encoding the asymptotic symmetries of 4D flat space. We derive these results by recasting 4D dynamics in terms of a convenient foliation of flat space into 3D Euclidean AdS and Lorentzian dS geometries. Tree-level scattering amplitudes take the form of Witten diagrams for a continuum of (A)dS modes, which are in turn equivalent to CFT correlators via the (A)dS/CFT dictionary. The Ward identities for the 2D conserved currents are dual to 4D soft theorems, while the bulk-boundary propagators of massless (A)dS modes are superpositions of the leading and subleading Weinberg soft factors of gauge theory and gravity. In general, the massless (A)dS modes are 3D Chern-Simons gauge fields describing the soft, single helicity sectors of 4D gauge theory and gravity. Consistent with the topological nature of Chern-Simons theory, Aharonov-Bohm effects record the “tracks” of hard particles in the soft radiation, leading to a simple characterization of gauge and gravitational memories. Soft particle exchanges between hard processes define the Kac-Moody level and Virasoro central charge, which are thereby related to the 4D gauge coupling and gravitational strength in units of an infrared cutoff. Finally, we discuss a toy model for black hole horizons via a restriction to the Rindler region.

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4D scattering amplitudes and asymptotic symmetries from 2D CFT

Received: December scattering amplitudes and asymptotic symmetries from 2D CFT College Park 0 U.S.A. 0 Open Access 0 c The Authors. 0 0 Pasadena , CA 91125 , U.S.A 1 Department of Physics, University of Maryland 2 Walter Burke Institute for Theoretical Physics, California Institute of Technology We reformulate the scattering amplitudes of 4D gravity in the language of a 2D CFT on the celestial sphere. The resulting CFT structure exhibits an OPE constructed from 4D collinear singularities, as well as in nite-dimensional Kac-Moody and Virasoro algebras encoding the asymptotic symmetries of 4D We derive these results by recasting 4D dynamics in terms of a convenient foliation of at space into 3D Euclidean AdS and Lorentzian dS geometries. Tree-level scattering amplitudes take the form of Witten diagrams for a continuum of (A)dS modes, which are in turn equivalent to CFT correlators via the (A)dS/CFT dictionary. The Ward identities for the 2D conserved currents are dual to 4D soft theorems, while the bulk-boundary propagators of massless (A)dS modes are superpositions of the leading and subleading Weinberg soft factors of gauge theory and gravity. In general, the massless (A)dS modes are 3D Chern-Simons gauge elds describing the soft, single helicity sectors of 4D gauge theory and gravity. Consistent with the topological nature of Chern-Simons theory, AharonovBohm e ects record the \tracks" of hard particles in the soft radiation, leading to a simple characterization of gauge and gravitational memories. Soft particle exchanges between hard processes de ne the Kac-Moody level and Virasoro central charge, which are thereby related to the 4D gauge coupling and gravitational strength in units of an infrared cuto . Finally, we discuss a toy model for black hole horizons via a restriction to the Rindler region. AdS-CFT Correspondence; Conformal Field Theory; Scattering Amplitudes 1 Introduction 2 Setup 2.1 Boundary coordinates Mode expansion from Milne4 to AdS3 Scaling dimensions from AdS3=CFT2 Witten diagrams in AdS3 Bulk-bulk propagator Bulk-boundary propagator Continuation from Milne4 to Mink4 Mink4 scattering amplitudes as CFT2 correlators Conserved currents of CFT2 Mink4 soft theorems as CFT2 Ward identities Equivalence of Milne4 and Mink4 soft limits Kac-Moody algebra of CFT2 Chern-Simons theory and multiple soft emission Abelian Chern-Simons theory Non-Abelian Chern-Simons theory Locating Chern-Simons theory in Mink4 Wess-Zumino-Witten model and multiple soft emission 3.10 Relation to memory e ects 3.10.1 Chern-Simons memory and the Aharonov-Bohm e ect 3.10.2 Chern-Simons level from internal soft exchange 3.11 Toy model for a black hole horizon Stress tensor of CFT2 Bulk-boundary propagator for AdS3 graviton Ward identity for CFT2 stress tensor Relationship to subleading soft theorems in Mink4 Virasoro algebra of CFT2 Chern-Simons theory and multiple soft emission Equivalence to AdS3 gravity Relation to asymptotic symmetries From super-rotations to super-translations in Mink4 Chern-Simons theory for super-translations? Future directions The AdS/CFT correspondence [1{7] has revealed profound insights into the dualities equating theories with and without gravity. As an explicit formalism, it has also given teeth to the powerful notion of holography, fueling concrete progress on longstanding puzzles in an array of subjects, ranging from black hole physics to strongly coupled dynamics. Still, AdS/CFT professes the limits of its own applicability: the entire construction rests pivotally on the infrastructure of warped geometry. In this paper, we explore a potential strategy for channeling the power of AdS/CFT into 4D Minkowski spacetime. This ambitious goal has a long history [8{16], typically with a focus on AdS/CFT in the limit of in nite AdS radius. Here we follow a di erent path, in line with the seminal work of [17, 18]. The crux of our approach is to foliate Minkowski spacetime into a family of warped 3D slices for which the methodology of AdS/CFT is applicable, recasting the dynamics of 4D at space into the grammar of a 2D CFT.1 We derive the central objects of this conjectured 2D CFT | namely the conserved currents and stress tensor | and show how the corresponding Kac-Moody and Virasoro algebras beautifully encode the asymptotic symmetries of 4D gauge theory [20{24] and gravity [25{27]. Our results give a uni ed explanation for the deep connections recently discovered [20{24, 28{32] between asymptotic symmetries and 4D soft theorems [33{37], allowing us to extend and understand these results further. As we will see, the 2D current algebras are dual to 3D Chern-Simons (CS) gauge elds that describe soft elds in 4D, and for which the phenomena of gauge [38{40] and gravitational \memories" [41{45] take the form of abelian and non-abelian Aharonov-Bohm e ects [46{48]. Let us now discuss our results in more detail. In section 2.1, we set the stage by de ning a convenient set of coordinates for 4D Minkowski spacetime (Mink4). These coordinates are formally anchored to a xed origin [17, 18, 49{51] intuitively representing the location of a hard scattering process. In turn, this choice naturally divides Mink4 into two regions: the 4D Milne spacetimes (Milne4) past and future time-like separated from the origin, and the 4D spherical Rindler spacetime (Rind4) space-like separated from the origin. We then choose coordinates in which Milne4 and Rind4 are foliated into slices at a distance from the origin, or equivalently at xed Milne time and Rindler radius, respectively. Each Milne slice is equivalent to 3D Euclidean anti-de Sitter space (AdS3). While this geometry is purely spatial from the 4D viewpoint, we will for notational convenience 1See [19] and references therein for a handy review of 2D CFT. space-like separated from the origin, respectively. Each region is then foliated into a family of warped slices, each at a xed proper distance from the origin. refer to it as AdS3 with the Euclidean signature implied. Similarly, each Rindler slice is equivalent to Lorentzian de Sitter (dS3) spacetime. In section 2.2, we show how the corresponding AdS3 and dS3 boundaries (@AdS3 and @dS3) de ne a 2D celestial sphere at null in nity | the natural home of massless asymptotic states. By choosing the analog of Poincare patch coordinates on the warped slices, we nd that the celestial sphere is labeled by complex variables (z; z) that coincide with the projective spinor helicity variables frequently used in the study of scattering amplitudes. The geometry of our setup is depicted in gure 1, and our basic approach is outlined in section 2.3. bearing in mind that the underlying spacetime is actually at [17, 18]. To do so, in sections 3.1 and 3.2 we apply separation of variables to decompose all the degrees of freedom in Milne4 into \harmonics" in Milne time, yielding a continuous spectrum of \massive" AdS3 elds. Here the AdS3 \mass" of each eld is simply its Milne energy.2 In section 3.3 we go on to show that the Witten diagrams of AdS3 elds are precisely equal to at space scattering amplitudes in Milne4, albeit with a modi ed prescription for LSZ reduction correspondence o ers a formalism to recast these scattering amplitudes as correlators of a certain CFT2 living on the celestial sphere. The operator product expansion corresponds to singularities in (z; z) arising from collinear limits in the angular directions. 2This energy is in general not conserved in the \expanding Universe" de ned by Milne spacetime, but it will be in a number of Weyl invariant theories of interest. embedding space. Here the mechanics of this continuation, as well as our calculations in general, are greatly simpli ed by employing the elegant embedding formalism of [56{60]. Notably, the appearance of dS3 suggests that the underlying CFT2 is non-unitary, as we see in detail. Putting it all together in section 3.5, we are then able to extend the mapping between 4D scattering amplitudes and 2D correlators to all of Minkowski spacetime. A natural question now arises: which 4D scattering amplitudes are dual to the 2D correlators of conserved currents? For scattering amplitudes in the Milne region, the Witten diagrams for these correlators will involve massless AdS3 elds. According to our decomposition into Milne harmonics, these massless modes have vanishing Milne energy, and thus correspond to the Milne soft limit of particles in the 4D scattering amplitude. In the case of gauge theory, we show in section 3.6 that the Milne soft limit coincides precisely with the usual soft limit taken with respect to Minkowski energy. As a result, the Ward identity for a conserved current in 2D is literally equal to the leading Weinberg soft theorem for gauge bosons in 4D, which we show explicitly for abelian gauge theory with matter as well as Yang-Mills (YM) theory. We thereby conclude that the conserved currents of the CFT2 automatically guides us to identify 4D soft limits with 2D conserved currents. Afterwards, in section 3.7 we show how the existence of a 2D holomorphic conserved current relates to the presence of an in nite-dimensional Kac-Moody algebra.3 Next, we go on to construct the explicit AdS3 dual of the CFT2 for the current algebra subsector. In section 3.8, we show that soft gauge bosons of a single helicity comprise a 3D topological CS gauge theory in AdS3 whose dual is the 2D chiral Wess-Zumino-Witten (WZW) model [63{66] discussed in section 3.9. As is well-known, this theory is a 2D CFT imbued with an in nite-dimensional Kac-Moody algebra. We show explicitly how hard particles in 4D decompose into massive 3D matter elds that source the CS gauge elds. Afterwards, we discuss the Kac-Moody level kCS and its connection to internal exchange of soft gauge bosons. Our results suggest that the level is related to the 4D YM gauge coupling via kCS 1=gY2M. We also show in section 3.10 how the topological nature of CS theories re ects the remarkable phenomenon of 4D gauge \memory" [38{40] in which soft elds record the passage of hard particles carrying conserved charges through speci c angular regions on the celestial sphere. In our formulation, these memory e ects are naturally encoded as abelian and non-abelian Aharonov-Bohm phases from the encircling of hard particle \tracks" by CS gauge elds. Interestingly, ref. [67] proposed that gauge and gravitational memories have the potential to encode copious \soft hair" on black hole horizons, o ering new avenues for understanding the information paradox, as reviewed in [68]. While black hole physics is not the primary focus of this work, our formalism does give a natural framework to study a toy 3Such a structure was observed long ago in amplitudes [61], serving as inspiration for the twistor model for black hole horizons which we present in section 3.11. In particular, by excising the Milne regions of spacetime, we are left with a Rindler spacetime that describes a family of radially accelerating observers. We nd that the CFT2 structure extends to include the early and late time wavefunction at the Rindler horizon. In particular, the 2D conserved currents are dual to CS soft elds that record the insertion points of hard particles that puncture the horizon and that escape to null in nity. In a parallel analysis for gravity, we show in section 4.1 that the Ward identity for the 2D stress tensor is an angular convolution of the subleading Weinberg soft theorem for gravitons in 4D. As for any CFT2, this theory is equipped with an in nite-dimensional Virasoro algebra that we discuss in section 4.2. Since the global SL(2; C) subgroup is nothing but the 4D Lorentz group, these Virasoro symmetries are aptly identi ed as the \superrotations" of the extended BMS algebra of asymptotic symmetries in 4D at space [25{27]. We then consider the case of subleading soft gravitons and the CFT2 stress tensor in section 4.3, arguing that the dual theory is simply AdS3 gravity, which famously is equivalent to a CS theory in 3D [69, 70]. Afterwards, we go on to discuss the connections between 4D gravitational memory, and the Virasoro algebra. While the value of the Virasoro central charge c is subtle, our physical picture suggests that c m2PlLI2R, where mPl is the 4D Planck scale and LIR is an infrared cuto . We then utilize the extended BMS algebra [71] to derive the CFT2 Ward identity associated with \super-translations" [25, 26], and we con rm that they correspond to the leading Weinberg soft theorem for gravitons [28, 29]. Finally, let us pause to orient our results within the grander ambitions of constructing a holographic dual to at space. Our central results rely crucially on the soft limit in 4D, wherein lie the hallmarks of 2D CFT. At the same time, a holographic dual to will necessarily describe all 4D dynamics, including the soft regime. Hence, our results imply that the soft limit of any such dual will be described by a CFT. In this sense, the CFT structure derived in this paper should be interpreted as a stringent constraint on any holographic dual to at space. Note added: during the nal stages of preparation for this paper, ref. [72] appeared, also deriving a 2D stress tensor for 4D single soft graviton emission. As outlined in the introduction, our essential strategy is to import the holographic correspondence into at space by reinterpreting Mink4 as the embedding space for a family of AdS3 slices [17, 18]. To accomplish this, we foliate Mink4 into a set of warped geomeand new facts about at space scattering amplitudes. We now de ne bulk and boundary coordinates natural to achieve this mapping. Bulk coordinates ds2Mink4 = and labeled by Greek indices ( ; ; : : :) hereafter. As outlined in the introduction, it will be convenient to organize spacetime points in Minkowski space according to their proper distance from the origin. This partitions at space into Milne and Rindler regions that are time-like and space-like separated from the origin. We foliate the 4D Milne region into hyperbolic slices of a xed proper distance from the is the Milne time coordinate. Together with the remaining spatial directions, de nes a set of 4D hyperbolic Milne coordinates, x2 = Y I = ( ; ; z; z); x0 = x3 = x1 + ix2 = ix2 = The domain for each Milne coordinate is ; 2 R and z; z 2 C. The regions < 0 correspond to the two halves of Milne4 | that is, the future and past Milne regions circumscribed by the future and past lightcones of the origin, respectively. So depending on the sign of , the ! +1 limit corresponds to either the asymptotic past or the asymptotic future. On the other hand, the 1 limit corresponds to the x2 = 0 boundary dividing the Milne and Rindler regions. In the context of a standalone Rindler spacetime, this boundary is known as the Rindler horizon.4 In the current setup, however, this horizon is a coordinate artifact simply because the underlying Minkowski space seamlessly joins the Milne and Rindler regions. Last but not least, (z; z) denote complex stereographic coordinates on the celestial sphere. Note that the physical angles on the sky labeled by (z; z) are antipodally identi ed for < 0, due to the diametric mapping between celestial spheres in the asymptotic past and the asymptotic future. By construction, the Milne coordinates are de ned so that Milne4 decomposes into a family of Euclidean AdS3 geometries, ds2Milne4 = GIJ (Y )dY I dY J = e2 Each slice at xed describes a 3D geometry equivalent to Euclidean AdS3 spacetime in Poincare patch coordinates [4], so ds2AdS3 = gij (y)dyidyj = 4More precisely, we are considering a spherical rather than the standard planar Rindler region reviewed denoted by upper-case Latin indices (I; J; : : :) hereafter. The Milne coordinates Y I are related to the Cartesian coordinates x according to From eq. (2.6) it is obvious that corresponds to the radial coordinate of AdS3 and ! 0 limit de nes the boundary @AdS3. Interpolating between the past and future Milne regions corresponds to an analytic continuation of the AdS3 radius to both positive and negative values. A similar analysis applies to the 4D Rindler region, which we foliate with respect to x2 = e2 ; where is now the Rindler radial coordinate. Like before, we can de ne hyperbolic Rindler coordinates, Y I = ( ; ; z; z), with the associated metric, ds2Rind4 = GIJ (Y )dY I dY J = e2 where lower-case Latin indices (i; j; : : :) denote AdS3 coordinates, yi = ( ; z; z); ds2dS3 = gij (y)dyidyj = is the conformal time of dS3. Boundary coordinates Given a hyperbolic foliation of Minkowski space, it is then natural to consider the spacetime boundary associated with each warped slice. To be concrete, let us focus here on Milne4, although a similar story will apply to Rind4. Using the Milne coordinates in eq. (2.4), we express an arbitrary spacetime point in x = e where we have de ned the null vectors, k = (1 + zz; z + z; iz + iz; 1 + zz) q = In terms of the celestial sphere, k is a vector pointing in the (z; z) direction while q is a reference vector pointing at complex in nity. Of course, while q describes a certain physical angle on the sky, this is a coordinate artifact without any physical signi cance. Given a null vector k it is natural to de ne polarization vectors, where and correspond to (+) and ( ) helicity states, respectively. As usual, the helicity sum over products of polarization vectors yields a projector onto physical states, where qk = and the reference vector q are compactly expressed in terms derivatives of k , The above expressions will be quite useful for manipulating expressions later on. To go to the boundary of AdS3 we take the limit of vanishing radial coordinate, According to eq. (2.11), any spacetime point at the boundary approaches a null vector, = @zk = @zk q = @z@zk : so @AdS3 is the natural arena for describing massless degrees of freedom. To appreciate the signi cance of this, recall that the in and out states of a scattering amplitude are inserted in the asymptotic past and future, de ned by this implies that null trajectories at ! +1. For massless particles, ! +1 should approach ! 0 so that asymptotic states originate at @AdS3 in the far past or terminate at @AdS3 in the far future. Said more precisely, @AdS3 is none other than past and future null in nity restricted to the Milne region.5 Hence, @AdS3 is a natural asymptotic boundary associated with the scattering of Finally, let us comment on the unexpected connection between our coordinates and the spinor helicity formalism commonly used in the study of scattering amplitudes. In particular, while the speci c form of k in eq. (2.12) was rigidly dictated by the choice of Poincare patch coordinates on AdS3, it also happens to be that are projective spinors, = (z; 1) = (z; 1); in a normalization where tr( ) = =2. Here are de ned modulo rescaling, i.e. modulo the energy of the associated momentum. This projective property implies that the only invariant kinematic data stored in Meanwhile, the reference vector q can also be expressed in spinor helicity form, k = q = 5Past and future null in nity in the Rindler region is contained in the boundary of dS3. are reference spinors, and the polarization vectors take the simple form, = (1; 0) = (1; 0); Thus, our hyperbolic foliation of Minkowski space has induced a coordinate system on the boundary that coincides with projective spinor helicity variables in a gauge speci ed by a particular set of reference spinors. As usual, we can combine spinors into Lorentz invariant angle and square brackets, h12i = 1 2 = z1 [12] = 1 _ 2 _ = z1 Meanwhile, the invariant mass of two null vectors, (k1 + k2)2 = h12i[12] = jz1 is the natural distance between points on the celestial sphere. As is familiar from the context of scattering amplitudes, expressions typically undergo drastic simpli cations when expressed in terms of spinor helicity variables. For example, the celebrated Parke-Taylor formula for the color-stripped MHV amplitude in non-abelian gauge theory is AnMHV = h12ih23i : : : hn1i Here the collinear singularities are manifest in the form of zi zi+1 poles in the denominator. More generally, since projective spinors only carry angular information, they are useful for exposing the collinear behavior of expressions. So far we have simply de ned a convenient representation of 4D Minkowski space as Milne and Rindler regions foliated into warped 3D slices. While at last we appear poised to apply slices | to which should we apply the holographic correspondence? After all, each value of corresponds to a distinct AdS3 geometry, each with a di erent curvature and position in Milne4. Even stranger, the bulk dynamics of Mink4 will in general intersect all foliations of both Milne4 and Rind4. The resolution to this puzzle is rather straightforward | and ubiquitous in more confactorizable geometry, AdS M, where M is a compact manifold. In such circumstances, the appropriate course of action is to Kaluza-Klein (KK) reduce the degrees of freedom along the compact directions of M. This generates a tower of KK modes in AdS to which the standard AdS/CFT dictionary should then be applied. In a slightly more complicated scenario, the spacetime is a warped product of AdS and M, where the AdS radius varies from point to point in M. Here too, KK reduction to AdS | with some ducial radius of curvature | can be performed, again resulting in a tower of KK modes. Something very similar occurs in our setup because Milne4 is simply a warped product of AdS3 and R , the real line parameterizing Milne time. Here \KK reduction" corresponds to a decomposition of elds in Milne4 into modes in Milne time which are in turn AdS3 elds via separation of variables. Each mode is then interpreted as a separate particle residing in the dimensionally reduced AdS3. However, unlike the usual KK scenario, where the spectrum of particles is discrete, the non-compactness of R induces a continuous \spectrum" of AdS3 modes. As we will see later, an e ective \compacti cation" [22] occurs when we consider the soft limit, which is the analog of projecting onto zero modes in the standard Kaluza-Klein procedure. In the subsequent sections we derive this mode decomposition for scalar and gauge theories in the Milne region. We consider theories that exhibit classical Weyl invariance, permitting Milne4 to be recast as a nicely factorized geometry, AdS3 R , rather than a warped product. In this case the mode decomposition is especially simple because Milne energy is conserved. Note, however, that this is merely a technical convenience that is not essential for our main results. In particular, when we go on to consider the case of gravity, there will be no such Weyl invariance, but the reduction of Milne4 down to AdS3 modes is of course still possible. Armed with a reduction of Milne4 degrees of freedom down to AdS3, we then apply the We then show how the embedding formalism o ers a trivial continuation of these results from Milne4 into Rind4 and thus all of Mink4. Along the way, we will understand the 4D interpretation of familiar objects in the CFT2, including correlators, Ward identities, and Gauge theory Mode expansion from Milne4 to AdS3 As a simple warmup, consider the case of a massless interacting scalar eld in Minkowski space. For the sake of convenience, we focus on Weyl invariant theories, although as noted previously this is not a necessity. The simplest Weyl invariant action of a scalar is S = for now restricting to the contribution to the action from Milne4. An identical analysis will apply to Rind4, and later we will discuss at length how to glue these regions together. In eq. (3.1) the conformal coupling to the Ricci scalar has no dynamical e ect in at stress tensor for the scalar that ensures Weyl invariance. The Weyl transformation is given where the scalar transforms as GIJ ! GIJ = e 2 GIJ ; = e Due to the classical Weyl invariance of the theory, the metric decomposes into a factorizable R geometry with the associated metric, = GIJ dY I dY J = where ds2AdS3 is de ned in eq. (2.6). Since the action is Weyl invariant we obtain S = where R = 6 is the curvature of the GIJ metric. Given the factorizable geometry, it is natural to de ne a \Milne energy", ! = i@ ; E = i@0; which is by construction a Casimir invariant under the AdS3 isometries, or in the language of the dual CFT2, the global conformal group SL(2; C). This SL(2; C) is also the 4D Lorentz group acting on the Milne4 embedding space of AdS3. By contrast, the usual is of course not Lorentz invariant and thus not SL(2; C) invariant, and so is less useful in identifying the underlying CFT2 structure. Again, we emphasize here that the Weyl invariance of the scalar theory is an algebraic convenience that is not crucial for any of our nal conclusions. When Weyl invariance is broken, then the Milne energy simply is not We can now expand the scalar into harmonics in Milne time, (!) = where (!) are scalar elds in AdS3, analogous to the tower of KK modes that arise in conventional compacti cations. In terms of these elds, the linearized action becomes S0 = so a massless scalar eld in Milne4 decomposes into a tower of AdS3 scalars with m2 (!) = Curiously, the mass violates the 3D Breitenlohner-Freedman bound [74, 75] and is thus formally tachyonic in AdS3. In fact, as the Milne energy grows, the mass becomes more tachyonic simply because we have mode expanded in a time-like direction. While such pathologies ordinarily imply an unbounded from below Hamiltonian, one should realize here that the AdS3 theory is Euclidean and the true time direction actually lies outside the warped geometry. Next, let us proceed to the case of 4D gauge theory. We consider the YM action, S = again focusing on contributions from the Milne region. Here FIJ is the Lie algebra-valued non-abelian gauge eld strength. Under a Weyl transformation, the metric transforms according to eq. (3.2), while the gauge eld is left invariant, Due to the classical Weyl invariance of 4D YM theory, this transformation leaves the action S = As before, the Weyl invariance of the action is a convenience whose main purpose is to simplify some of the algebra. A = 0, we rewrite the linearized action as S0 = where fij = @iaj @j ai is the linearized eld strength associated with the Milne modes, From eq. (3.14) we see that the ai(!) are Proca vector elds in AdS3 with mass ai(!) = m2a(!) = elds are formally tachyonic since we have mode expanded in the time-like Milne direction. In summary, we nd that a massless vector in Milne4 decomposes into a continuous tower of massive Proca vector elds in AdS3. Scaling dimensions from AdS3=CFT2 According to the standard holographic dictionary, each eld in AdS3 is dual to a CFT2 primary operator with scaling dimension dictated by the corresponding AdS3 mass. From eq. (3.10) and eq. (3.16), we deduce that the scaling dimensions for scalar and vector pri2) = m2 (!) = 1)2 = m2a(!) = !2. Both equations imply the following relationship between the scaling dimension and the Milne energy, (!) = 1 Since unitary CFTs and their Wick-rotated Euclidean versions have real scaling dimensions, the CFT encountered here is formally non-unitary. This is true despite the manifest unitarity of the underlying 4D dynamics. With the mode decomposition just discussed, it is a tedious but straightforward exercise to derive an explicit action for the tower of AdS3 modes descended from Milne4. From this these Witten diagrams are equivalent to correlators of a certain CFT2. As we will argue here and in subsequent sections, these Witten diagrams and correlators are also equal to scattering amplitudes in Mink4. A priori, such a correspondence is quite natural. First of all, tree-level Witten diagrams and scattering amplitudes both describe a classical minimization problem | i.e. nding the saddle point of the action subject to a particular set of boundary conditions. Second, the CFT2 resides on the @AdS3 boundary, which at ! +1 houses massless asymptotic in and out states. In any case, we will derive an explicit mapping between the basic components of Witten diagrams and scattering amplitudes. The former are comprised of interaction vertices, bulk-bulk propagators, and bulk-boundary propagators, while the latter are comprised of interaction vertices, internal propagators, and a prescription for LSZ reduction. Let us analyze each of these elements in turn. Interaction vertices To compute the interaction vertices of the AdS3 theory we simply express the interactions in Milne4 in terms of the mode decomposition into massive AdS3 elds. For example, the quartic self-interaction of the scalar eld becomes Sint = d!1d!2d!3d!4 (!1) (!2) (!3) (!4) (!1 + !2 + !3 + !4); so interactions in the bulk of Milne4 translate into interactions among massive scalars in AdS3. Due to the Weyl invariance of the original scalar theory, these interactions conserve It is then clear that the interaction vertices of 3D Witten diagrams are equivalent to at space Feynman diagrams modulo a choice of coordinates | that is, Milne versus Minkowski coordinates, respectively. While these Witten diagram interactions typically involve complicated interactions among many AdS3 elds, this is just a repackaging of standard Feynman vertices. Bulk-bulk propagator In this section we show that the bulk-bulk propagators of Milne harmonics in AdS3 are simply a repackaging of Feynman propagators in Mink4. To simplify our discussion, let us again revisit the case of the massless scalar eld, although a parallel discussion holds for gauge theory but with the extra complication of gauge xing. Consider the Feynman propagator for a massless scalar eld in at space, G( ; y; 0; y0)Mink4 = = e 0 where ( ; y) and ( 0; y0) are points in the Milne region. Here we have de ned Mink4 = rI r AdS3 = riri; to be the d'Lambertian in Mink4 and the Laplacian in AdS3, respectively. This expression is manifestly of the form of the AdS3 propagator with e factors inserted to account for the non-trivial Weyl weight of the scalar eld. Indeed, by applying the Weyl transformation and decomposing into Milne modes, we obtain the AdS3 propagator for a scalar, G(!; y; y0)AdS3 = which automatically satis es the wave equation for a scalar in AdS3, (riri + 1 + !2)G(!; y; y0)AdS3 = i 3(y; y0): Hence, the Feynman propagator is a particular convolution over a tower of AdS3 propagausual i prescription, Of course, the above statements are purely formal until the di erential operator inverses are properly de ned by an i prescription. The Minkowski propagator takes the G( ; y; 0; y0)Mink4 = which selects the Minkowski vacuum as the ground state of the theory. This is, however, not the natural vacuum of the Weyl-transformed geometry, AdS3 R , which is instead the conformal vacuum corresponding to the ground state with respect to the Milne Hamiltonian, translations. In order to match the propagator of the Minkowski vacuum we must choose the thermal propagator in AdS3 R [73]. Thermality arises from the entanglement between the Milne and Rindler regions of Minkowski spacetime across the Rindler horizon bulk-bulk propagators in AdS3. Note that thermality does not break the SL(2; C) Lorentz symmetries, since these act only on the AdS3 coordinates and not the Milne time or energy. A similar story holds for gauge elds. Going to Milne temporal gauge, the Mink4 gauge propagator can be expressed as a convolution over massive AdS3 Proca propagators. These propagators satisfy the Proca wave equation sourced by a delta function, rirj + !2 ij )Gjl(!; y; y0)AdS3 = i il 3(y; y0); where we have Fourier transformed to Milne harmonics. Bulk-boundary propagator We have now veri ed that the bulk interaction vertices and bulk-bulk propagators of Witten diagrams in AdS3 are simply Feynman diagrammatic elements in the Milne4 embedding nal step in matching Witten diagrams to scattering amplitudes is to match their respective boundary conditions. For Witten diagrams, the external lines are AdS3 bulk-boundary propagators. For scattering amplitudes, the external lines are xed by LSZ reduction to be solutions of the Mink4 free particle equations of motion | taken usually to be plane waves. Here we derive a concrete relationship between the bulk-boundary propagators and LSZ reduction. To begin, let us compute the bulk-boundary propagator for primary elds of scaling . At this point it will be convenient to employ the elegant embedding formalism of [60], which derived formulas for the bulk-boundary propagator in terms of a embedding space of one higher dimension. Ordinarily, AdS is considered physical while the at embedding space is an abstraction devised to simplify the bookkeeping of curved spacetime. Here the scenario is completely reversed: at space is physical while AdS is the abstraction introduced in order to recast at space dynamics into the language of CFT. In the embedding formalism [60], the bulk-boundary propagator for a scalar primary is Since we have lifted from AdS3 to Mink4, the right-hand side actually depends on 4D quantities. Speci cally, the four-vector x labels a point in Mink4 while the four-vector k labels a point (z; z) on the boundary of AdS3 according to eq. (2.12). Already, we see an elegant subtlety that arises in the embedding formalism: each 1. In Milne coordinates, this corresponds to the constraint = 0. We can, however, \lift" the bulkboundary propagators from AdS3 to Mink4 by simply dropping this constraint, yielding a bulk-boundary propagator with an additional dependent factor, e . Combined with an extra factor of e for the Weyl weight of a scalar eld, this generates a net phase e ( 1) = e i! from the de nition of in eq. (3.17). We immediately recognize this as the phase factor that accompanies the Fourier transform between dependent elds in AdS3 and ! dependent Milne harmonics. That is, the lifted propagators can be used to compute the boundary correlators of modes in AdS3 in terms of boundary correlators of 4D states in R . The fact that the bulk-boundary propagators satisfy the free particle equations of motion in AdS3 translates to the fact that the Weyl-transformed lifted propagators satisfy the free particle equations of motion in AdS3 R via separation of variables. In turn, this implies that the embedding formalism bulk-boundary propagator in eq. (3.25) satis es the equations of motion in Mink4. This fact is straightforwardly checked by direct computation. Next, consider the bulk-boundary propagator for a vector primary, Ki . This object is fundamentally a bi-vector since it characterizes propagation of a vector disturbance from the @AdS3 boundary into the bulk of AdS3. While the 3D bulk vector index is manifest, the 2D boundary vector index is suppressed | implicitly taken here to be either the z or z component. As for the scalar, we can lift the AdS3 bulk-boundary propagator to = 0. Going to Minkowski coordinates, we obtain KI = where we have chosen the z component of the boundary vector. Here the dependence on boundary coordinates (z; z) enters through k and according to eq. (2.12) and eq. (2.13). Had we instead chosen the z component of the boundary vector, we would have obtained the same expression as eq. (3.26) except with instead of . Continuation from Milne4 to Mink4 Until now, the ingredients of our discussion | interaction vertices, bulk-bulk propagators, and bulk-boundary propagators | have all been restricted to Milne region time-like separated from the origin. However, it is clear that scattering processes in general will also involve the Rindler region space-like separated from the origin. As we will see, this is not a problem because the Milne diagrammatic components | written in terms of at space coordinates via the embedding formalism | can be trivially continued to the Rindler region and thus all of Minkowski spacetime. To be concrete, recall the foliation of the Rindler region in eq. (2.8) and eq. (2.9). Each de nes a Lorentzian dS3 spacetime. In Rind4, boundary correlators correspond to Witten diagrams of dS3 elds descended from a mode decomposition with respect Rind4 are given precisely by eq. (3.25) and eq. (3.26), except continued to the full Mink4 region for any value of x2. So the embedding formalism gives a perfect prescription for continuation from Milne to Rindler. One can also think of this as a simple analytic continuation of the original AdS3 theory into dS3, which shares the same SL(2; C) Lorentz isometries. This result implies that Mink4 scattering amplitudes | properly LSZ-reduced on bulkboundary propagators on both the Milne and Rindler regions | are equal to a 3D Witten diagrams for Milne and Rindler harmonics which splice together boundary correlators in AdS3 and dS3. Using these continued Witten diagrams, we can then de ne a set of given the Euclidean signature of AdS3 and the Lorentzian signature of dS3. As a consequence, our proposed correspondence between Mink4 and CFT2 is subtle. While the Minkowski theory is unitary, the CFT2 is not unitary in any familiar sense | a fact which is evident from the appearance of complex scaling dimensions in eq. (3.17). This space unitarity is encoded within a non-unitary CFT obviously deserves further study. Mink4 scattering amplitudes as CFT2 correlators Assembling the various diagrammatic ingredients, we see that Witten diagrams for the elds descended from the mode decomposition of Mink4 are equal to 4D scattering amplitudes | albeit with a modi ed prescription for LSZ reduction in which the usual external wavepackets of xed momentum are replaced with the lifted bulk-boundary propagators of eq. (3.25) and eq. (3.26). These alternative \wavepackets" may seem unfamiliar, but crucially, they can be expressed as superpositions of on-shell plane waves. For the scalar eld this is straightforward, since the bulk-boundary propagator in eq. (3.25) can be expressed as a Mellin transform of plane waves [17], where " is an in nitesimal regulator. Here the right-hand side is manifestly a superposition of on-shell plane waves, eiskx, since k2 = 0. Something similar happens for the gauge eld since Using the simple observation that k @z( ) = @z(k ) ( ), we see that eq. (3.27) and eq. (3.28) imply that K is a superposition of on-shell plane waves, superposition of pure gauge transformations, k eiskx. In this way, we have shown that every Witten diagram can be written as a superposition of on-shell scattering amplitudes in Mink4, or equivalently as a single scattering amplitude with a modi ed LSZ-reduction to certain bulk-boundary wavepackets. By the (A)dS/CFT dictionary, this implies that the latter are equivalent to Euclidean correlators of a CFT2 on the @(A)dS3 boundaries, which together form the entirety of past and future null in nity. Concretely, this implies the equivalence of correlators and scattering amplitudes, n (zn; zn)i = A(K 1 (z1; z1); : : : ; K n (zn; zn)) = houtjini; where here we have restricted to scalar operators for simplicity, but the obvious generalization to higher spin applies. In eq. (3.29) the quantity A denotes a scattering amplitude with a modi ed LSZ-reduction replacing the usual plane waves with the lifted bulk-boundary propagators K i (zi; zi) corresponding to the boundary operators O i (zi; zi). The associinvariant in 4D, for example as in massless gauge theory at tree level, then Pn i=1 !i = 0. The boundary operators are naturally divided into two types, Oin and Oout, depending on sign of the Minkowski energy E > 0 or E < 0, corresponding to scattering states that are incoming or outgoing, respectively. This equivalence of correlators and scattering amplitudes is depicted in gure 2. Conserved currents of CFT2 In eq. (3.29), we derived an explicit holographic correspondence between scattering amplitudes in Mink4 and correlators of a certain CFT2. For gauge elds, the associated massive AdS3 modes are dual to non-conserved currents in the CFT2 while the massless AdS3 modes are dual to conserved currents in the CFT2. Since the mass of an AdS3 vector is proportional to its Milne energy by eq. (3.16), we can study the massless case by taking primary operator, this corresponds to for current conservation in the CFT2. Scattering Amplitude Correlator W (x) = houtjJ (x)jini; hO(zi, z¯i)j(z1)j(z2) · · · j(zn)O(zj, z¯j) · · · i multiple softhib1os1o2ni ·g·a·uhgne em1nissinojni and multiple conser(vzeid czu1r)r(ezn1t inzs2e)rtion(z.n 1 ih ih · · · To start, consider the bulk-boundary propagator for a massless AdS3 vector, obtained by setting constructed from boundary data, = k = k Note that x K = K is actually a total derivative with respect to Mink4 coordinates, = @ This fact dovetails beautifully with the results of [20, 21, 23, 24], which argued that there is physical signi cance to large gauge transformations that do not vanish at the boundary of Mink4. As we will see, concrete calculations are vastly simpli ed using the pure gauge form of K . Mink4 soft theorems as CFT2 Ward identities Let us start with the simplest case of abelian gauge theory with arbitrary charged matter. We showed earlier that a Mink4 scattering amplitude with a Milne soft gauge boson can be expressed as a Witten diagram for a massless AdS3 vector eld, hj(z)O(z1; z1) O(zn; zn)i = d4x K (x)W (x): Here the left-hand side is a correlator involving the = 1 conserved current of the CFT2 and K is the bulk-boundary propagator for the massless vector in AdS3. The function represents the remaining contributions to the Witten diagram from bulk interactions, where J is the gauge current operator of 4D Minkowski spacetime inserted between scattering states. Here the in and out states are de ned according to the modi ed prescription for LSZ reduction shown in eq. (3.29). Inserting the pure gauge form of K in eq. (3.32) and integrating by parts, we obtain hj(z)O(z1; z1) O(zn; zn)i = By dropping total derivatives, we have implicitly assumed that W describes a charge con guration that vanishes on the boundary. Naively, this stipulation is inconsistent if the bulk process involves charged external particles that propagate to the asymptotic boundary. However, this need not be a contradiction, provided W is sourced by insertions of charged particles near but not quite on the boundary. Conservation of charge is e ectively violated wherever the external particles are inserted, so @ houtjJ (x)jini = Here i runs over all the particles in the scattering process, qi are their charges, and xi are their insertion points near the @AdS3 boundary. Crucially, we recall from eq. (2.11) that massless particles near the @AdS3 boundary are located at positions xi that are aligned with their associated on-shell momenta, ki. This is simply the statement that the positions of asymptotic states on the celestial sphere point in the same directions as their momenta. In any case, the upshot is that as i ! 0, we can substitute xi Plugging in eq. (3.32) and eq. (3.29), and replacing xi ki, we can trivially integrate the delta function to obtain hj(z)O(z1; z1) O(zn; zn)i = which is exactly the Weinberg soft factor for soft gauge boson emission [33]. Here it was important that we identi ed xi ki so that the resulting Weinberg soft factor depends on the on-shell momenta, ki. Later on, we will occasionally nd it useful to switch back and forth between the position and momentum basis for the hard particles. At the same time, this expression simpli es further because yielding the Ward identity for a 2D conserved current, hj(z)O(z1; z1) O(zn; zn)i = So eq. (3.37) is simultaneously the soft theorem in Mink4, the Witten diagram for a massless vector in AdS3, and the Ward identity for a conserved current in the CFT2. From this ki = kCS, suggesting that kCS 1=gY2M. electric eld. However, by the classical eld equations, this is proportional to @ Bi, which vanishes in the Milne soft limit. We thereby conclude that the AB phase and by extension the CS gauge eld is equivalent to the memory eld. Chern-Simons level from internal soft exchange We have just seen how the AB e ect in the 3D CS description for 4D soft emission encodes a velocity kick for charged particles that embodies the electromagnetic memory e ect. While electromagnetic memory is most simply measured with massive charged probes, an alternative approach would be to con gure a secondary hard process comprised massless charged particles that measure the soft emission from an initial scattering. In the CS theory, this corresponds to diagrams composed of disjoint charged currents connected only by the exchange of an internal CS gauge line, as depicted in gure 7. This requires a new element, as thus far we have only matched the external CS lines to external soft emission lines in 4D. Obviously, the exchange of an internal CS gauge eld in AdS3 is dual to a Mink4 scattering amplitude with an internal soft gauge boson exchange. Such an amplitude describes two hard processes connected by a soft internal gauge boson, so it only occurs at very special kinematics. Since this particle travels a great distance before it is reabsorbed, it can be assigned a helicity. The external soft emission and absorption processes studied earlier are then just sewn together as factorization channels of this composite process. Internal gauge exchange in CS is also important in because it encodes the CS level, kCS, re ecting quantum uctuations of the gauge eld. When the Lie algebra is normalized independently of the couplings of the gauge theory, the CS action reads while the action for YM theory in 4D is SCS = 3 AiAj Ak "ijk; SYM = Notably, the solutions to the classical CS and YM equations of motion do not depend on kCS nor gYM since these are prefactors of the action, and thus drop out of the homogenous eld equations. Said another way, at tree level these couplings can be reabsorbed into the de nition of ~. Hence, the gauge eld describing the soft external branches depicted in gure 5 are actually independent of these parameters. On the other hand, these variables do enter into diagrams with internal CS gauge lines, or equivalently Mink4 processes with intermediate soft gauge boson exchange. In CS perturbation theory [79, 80], the former comes the self-dual constraint, F gure 6, we nd that SYM = in agreement with [20] but not [22], which argued for a vanishing Kac-Moody level. This result can also be obtained from the following heuristic derivation. Substituting = iF~ , into the YM action in the regulated Milne region of ~ ) = ! +1. Matching this to the CS action, we verify eq. (3.91). which is a total derivative. In principle, this total derivative will integrate to all the boundaries of the regulated Milne4. However, due to our choice of Milne temporal gauge Thus, we again obtain the non-abelian CS action in eq. (3.89) where Ai is the gauge eld Toy model for a black hole horizon As recently discussed [67], it is interesting to understand in what sense asymptotic symmetries and the memory e ect constitute a new kind of \hair" in the presence of black hole horizons. While this paper has focused on uncovering a CFT2 structure underlying Mink4 scattering amplitudes, our strategy incidentally o ers a baby version of the black hole problem in the form of the Rindler horizon, say as seen by radially accelerating observers in the Rindler region. For such observers we can excise all of Mink4 spacetime that lies behind a \stretched" Rindler horizon, excluding the Milne regions altogether, as depicted in gure 8. The physical observables relevant to the remaining Rindler region are thermal correlators6 which encode the wavefunction describing the particles emitted to or from null in nity together with the stretched horizon. First, let us remind the reader of the Rind4 coordinates in eq. (2.9), where each hyperbolic slice at xed Rindler radius de nes a dS3 spacetime labeled by conformal time . As discussed earlier, the roles of in the Rindler region are swapped relative to the Milne region. So for any dS3 slice, the corresponding @dS3 boundary is de ned by the end of time limit ! 0. Meanwhile, ! +1 corresponds to null in nity, while 6Here \thermal" is with respect to dS3 time in static patch coordinates, as experienced by a Rindler early and late times on the stretched horizon. Therefore a correlator in Rind4 has the form of a Mink4 correlator, houtjO1 Here the in and out states label particles emitted from and to null in nity in the far past or future, respectively. Meanwhile, the operators Oi denote insertions of particle elds on the stretched horizon at early or late times. These operators are generic probes of the wavefunction of the stretched horizon. Despite the fact that we have restricted physical spacetime to the Rindler region outside the stretched horizon, we must compute this correlator using Minkowski Feynman diagrams in order to match the thermal Rindler correlators. Such diagrams will now consist of four ingredients: the three already discussed | interaction vertices, propagators, and LSZ wave packets | together with additional propagators running from the Oi inserted in the far past or future of the stretched horizon to interactions in the bulk of the Rindler region. Since the stretched horizon at stretch has Lorentzian dS3 geometry, these additional propagators describe a bulk point in dS3 and a boundary point on @dS3, so they are bulk-boundary propagators from this perspective. Therefore by the close analogy with our Milne manipulations, we see that the Rind4 correlators are boundary correlators in a dS3 theory which can be reinterpreted late time wavefunction of the Universe [54]. So the CFT2 describing the Rindler region is dual to the late time wavefunction of Rindler, up to and including the stretched horizon and given initial conditions for the wavefunction at early times. In this context, let us analyze the physics of the CS gauge eld and electromagnetic memory in the Rindler region. By the exact analog of eq. (3.75), we can locate the CS eld in Rind4 by taking the limit of soft Rindler momentum, so the CS eld corresponding to (+) helicity is Ai(y) = Ai+( Ai+( = stretch; y): The second term represents the component of the soft \memory" eld that remembers the hard charges that fall into the Rindler horizon. We see this explicitly because, retaining this component, the analog of our AB phase associated with a region R on null in nity of the Milne region in eq. (3.82), now reads in Rindler as ! 0; z) = where again the above expression is implicitly evaluated within a 2D correlator. Here Az is given by the two terms in eq. (3.93). We thereby conclude that the AB phase measures hard charges passing through an angular region R, regardless of whether those hard charges are falling into the horizon or are headed out to null in nity. If one measures the charges heading out to null in nity, the CS eld will encode information on where exactly the hard charges entered the horizon. This in some sense o ers a sharper form of \hair" [67] compared to the usual asymptotic electric eld of a black hole, which remembers the charge that has fallen into the horizon but without regard to the angle of entry. We have described the emergence of CFT structure in gauge theory amplitudes, but of course the hallmark of a true CFT is a 2D stress tensor. The Sugawara construction yields a stress tensor constructed from the 2D holomorphic currents dual to soft gauge elds, but this can only be a component of the full stress tensor since it does not account for hard particle dynamics. As usual in AdS/CFT, to nd the full stress tensor we must consider gravity, to which we now turn. Our aim will be to reframe many of the important aspects of 4D gravity in terms of the language of 2D CFT. We will follow the same basic strategy for gravity as for scalar and gauge theory, moving briskly through those aspects which are closely analogous and focusing on those which introduce major new considerations. The most important such consideration is that gravity in asymptotically at space is not Weyl invariant, since the 4D Einstein-Hilbert action, SEH = depends on the dimensionful Planck mass, mPl. For the sake of exposition, we will often restrict to the Milne region for explicit calculations, bearing in mind that we can straightforwardly continue into the Rindler region and thus all of Minkowski space via the embedding In any case, while the dynamics cannot be mapped into a factorizable geometry like R , this is merely a technical inconvenience. As in gauge theory, one can nevertheless apply a decomposition into AdS3 and dS3 modes, resulting in 3D Witten diagrams equivalent to 4D scattering amplitudes with a particular prescription for LSZ reduction onto bulk-boundary propagators. In this section, we derive a 2D stress tensor corresponding to soft gravitons in 4D. We will show that the Ward identity for the 2D stress tensor is a particular angular convolution of the subleading soft factor for graviton emission [37]. Notably, the subleading soft fact di ers from the leading factor in that it depends on the angular momentum of each external leg rather than the momentum. The pursuit of a 2D stress tensor will naturally lead us to the Virasoro algebra, which directly manifests the super-rotation [27] asymptotic symmetries of 4D Minkowski Commuting these with ordinary translations, we then derive the BMS supertranslations [25, 26]. This approach is anti-historical, but more natural from the holographic approach taken here. Bulk-boundary propagator for AdS3 graviton formalism, we write down the bulk-boundary propagator for hij lifted from 3D to 4D via = x Here, the normalization N (x) parameterizes an inherent ambiguity in the lift, arising because AdS3 lives on the constrained surface x 2 = 1. For gauge theory we sidestepped this ambiguity, since the underlying Weyl invariance implied that the dynamics are independent of the scale set by the constrained surface. However, there is no such invariance of 4D gravity due to the dimensionful gravitational constant, so we must nd an alternate way to identify N (x). Of course, N (x) should be chosen so K is a solution of the linearized Einstein's choice. The reason for this is that in standard AdS3 gravity the Virasoro symmetries arise as asymptotic symmetries of AdS3 encoded in solutions to the 3D Einstein's equations. Famously, all such solutions are pure gauge [81, 82], and are thus di eomorphisms of AdS3 itself. At the linearized level this is re ected in the fact that the bulk-boundary propagator rasoro symmetries as asymptotic symmetries of Mink4, we should look for a lift of the AdS3 bulk-boundary propagator that yields a pure linearized large di eomorphism in Mink4. A straightforward calculation shows that for the bulk-boundary propagator K the Milne4 is a warped product of AdS3 and R associated with a warp factor x which we now see is crucial to lift 3D di eomorphisms into a 4D di eomorphisms. Fixing = 2 is 2 = e2 , Since this is a pure di eomorphism, it can be written as = @ is de ned in eq. (3.31). This form for K will be quite useful for explicit Applying the logic of AdS/CFT, the bulk-boundary propagator for hij corresponds to the insertion of a local CFT2 stress tensor t(z) or its complex conjugate t(z). In the subsequent sections, we will see how the bulk-boundary propagator K relates to single and multiple soft graviton emission in 4D. Finally, let us comment on the curious fact the bulk-boundary propagator for gravity is proportional to the square of the bulk-boundary propagator for gauge theory, so = x2K K : arise from the KLT [83] relations and the closely related BCJ [78] relations. Given also the connection between BCJ and the soft limit [84], it is likely that the above equation is not an accident, and is perhaps a sign of some deeper underlying construction. Ward identity for CFT2 stress tensor Given the central role of the 2D stress tensor t(z), it is natural to ask about the 4D dual of this quantity. Repeating our strategy for gauge theory, we now calculate the Ward identity for the 2D stress tensor using AdS/CFT. To do so, we compute a correlator of the stress tensor via the associated Witten diagram, ht(z)O(z1; z1) O(zn; zn)i = is the bulk-boundary propagator in eq. (4.3) and W parameterizes the remainder of the Witten diagram, = houtjT computed as an insertion of the 4D stress tensor operator T inserted between in and out Substituting the pure gauge form of the bulk-boundary propagator in eq. (4.4), we ht(z)O(z1; z1) O(zn; zn)i = where in the second line we have shu ed around terms and performed an integration by parts, dropping boundary terms. Importantly, the expression sandwiched between in and out states is the relativistic angular momentum tensor. This quantity is conserved everywhere except at insertions associated with the external legs, so (x)jini = where Ji is the angular momentum of each external particle and xi is its insertion point near the boundary. As before, we substitute the position of the external particles inserted near the boundary i ! 0 with their corresponding momenta, so xi ki. As a result, the expression for the Ward identity will involve manifestly on-shell quantities. Plugging this substitution into the Ward identity, we obtain ht(z)O(z1; z1) O(zn; zn)i = O(zn; zn)i: (4.10) In the above equation, the angular momentum generator is implicitly de ned in momentum basis, so e.g. it acts on a hard scalar leg as = ki @ki The analogous expression for hard legs with spin has a simple representation in terms of spinor helicity variables. From eq. (4.10) we see directly the connection between the stress tensor in the CFT2 and rotations acting on the boundary of Mink4. This is not accidental, and as we will see later is a hint of the super-rotation asymptotic symmetries of 4D To compare this to the usual 2D stress tensor Ward identity, it is actually convenient to brie y revert to position space for the hard particles. To do so we send ki eq. (4.10) and eq. (4.11) and go to Milne coordinates. Taking the i ! 0 limit, we nd ht(z)O(z1; z1) where the conformal weight is hi = O(zn; zn)i; (4.12) for a 2D scalar operator dual to a hard 4D scalar particle. Up to an overall constant normalization, eq. (4.12) is none other than the Ward identity for the stress tensor of the CFT2. Of course, this analysis can be extended straightforwardly to include hard particles Relationship to subleading soft theorems in Mink4 Next, we derive the explicit relationship between the Ward identity for the 2D stress tensor and the soft graviton theorems. To do so, it will be convenient introduce an auxiliary operator t~(z; z) which is a do not assign independent physical import to this = 0 operator, which is why we refer to it as auxiliary. From the embedding formalism, the bulk-boundary propagator for t~(z; z) is Importantly, this bulk-boundary propagator is a pure linearized di eomorphism equal to = @ ~ + @ ~ ~ = Repeating our steps from before, calculate an arbitrary correlator involving t~(z; z), ht~(z; z)O(z1; z1) where the right-hand side is literally the subleading graviton soft factor [37]. While interesting, this observation is only useful because t~(z; z) happens to be directly related to t(z) by a handy integral transform in (z; z). Indeed, by comparing the de nitions of ~ in eq. (4.4) and eq. (4.15), respectively, we see that these quantities are related by the di erential equation, dropping unimportant numerical prefactors. Notably, the above equation is equivalent to the CFT2 equation @zt(z) @z3t~y(z; z), which when evaluated inside a correlator yields zero on both sides except for delta function support at the insertion points of hard operators. = 0, this zi), producing a delta function 2(z zi) from the identity in eq. (3.54). This is a non-trivial check that the structure of the subleading graviton soft theorem ensures conservation of the CFT2 stress tensor. In any case, we would like to solve the di erential equation in eq. (4.17) by constructing a formal anti-derivative, @z 1 = 1 Z dependence. Inserting this relation into the Ward identity for the stress tensor, we obtain our nal expression, ht(z)O(z1; z1) where k0, 0, and f 0 are functions of (z0; z0). This result says that the Ward identity for the 2D stress tensor is proportional to a particular angular integral over the subleading soft graviton factor. Physically, this corresponds to a particular superposition of soft graviton emission in all directions (z0; z0). Let us pause to discuss the peculiar integral structure of eq. (4.20). Naively, it is odd that the CFT2 stress tensor should be expressed as a non-local function in (z; z) but this was actually essential to maintain consistency between the 2D and 4D pictures. To see why, recall from eq. (4.12) that the canonical form of the 2D stress tensor Ward identity has manifest double and single poles in z. In turn, this OPE corresponds to collinear singularities in 4D, but graviton scattering amplitudes are famously free of such collinear singularities. Hence, the only way to square these apparently inconsistent statements is if the 2D stress tensor is actually a non-local function of the graviton scattering amplitude in (z; z), as eq. (4.20) clearly is. Only then is it possible for the singularity structure of the 2D stress tensor Ward identity to arise consistently from the analytic properties of graviton amplitudes. Virasoro algebra of CFT2 The Virasoro algebra places immense constraints on the structure of correlators in the CFT2. It is obviously of great interest to understand the implications of these constraints on the dual scattering amplitudes in Mink4. As we will see, the corresponding in nitedimensional Virasoro algebra in 2D has a direct connection to the asymptotic symmetries at space [27, 30]. in z, we obtain What is the action of the Virasoro generators on scattering amplitudes? To answer this, we revisit the 2D stress tensor Ward identity in eq. (4.10). Expanding the derivatives 2f 0 =3 @z0 f 0 =3 @z20 f 0 =3 For simplicity, consider the limit in which the soft graviton is collinear to a hard external leg located at z0 on the celestial sphere. A Laurent expansion of this expression around z = z0 yields ht(z)O(z0; z0) i = ht(z)O(z1; z1) O(zn; zn)i = ellipses denote non-singular contributions which originate from the other hard legs in the We can now compare eq. (4.22) directly to de nition of the Virasoro generators, t(z) = m= 1 zm+2 = : : : + of the SL(2; C) Virasoro generators, i(K2 + iJ2) i(K2 + iJ2) + (K1 + iJ1); up to a constant normalization factor. Here K3 and J3 denote the generators of J 0 corresponding to boosts and rotations around the axis of the hard particle, while K1;2 and J1;2 are those for the transverse directions. Since these generators only act on the collinear hard particle, they are e ectively local Lorentz transformations. Thus, the identi cation of the full Virasoro algebra as the algebra of super-rotations is indeed appropriate. This result o ers a physical interpretation for the action of t(z) on scattering amplitudes. The passage of collinear emitted soft gravitons induces a Lorentz transformation that acts locally on a hard leg. Operationally, this \jiggles" the hard particle in a way that displaces it relative to the direction of its original trajectory. This local Lorentz transformation has the same e ect as a net displacement of the detectors residing at the boundary Chern-Simons theory and multiple soft emission To understand multiple soft emissions in gravity, we proceed in parallel with our analysis for non-abelian gauge theory. Our aim is to describe the dynamics of multiple external soft gravitons that interact and merge in the gravitational analog of gure 4. As before, we can parameterize the dynamics of the entire soft branch with a graviton eld H at the juncture x with the hard process characterized by T (x). In the limit of vanishing gravitational coupling, H will approach a superposition of independent soft gravitons, each described by the bulk-boundary propagator K from eq. (4.3). Hence, the branch structure of soft gravitons is rooted in external legs connected through these bulk-boundary propagators. Said another way, the soft branch is simply the solution to the non-linear sourceless Einstein's equations with free- eld approximation given by K . Now consider a closely analogous situation for 3D Witten diagrams, where an AdS3 branch eld hij (y) similarly characterizes the web of soft gravitons merging before making contact with a hard source at y. Here hij can be treated as a perturbation of the background AdS3 metric gij de ned in eq. (2.6). The full metric in 3D is then g~ij = gij + hij ; where gij is the background AdS3 metric from eq. (2.6). Eq. (4.25) is a solution to Einstein's equations in AdS3 whose free eld asymptotics near @AdS3 are given by bulk-boundary propagators. Since all solutions to AdS3 gravity are pure di eomorphisms of AdS3 [81, 82], hij corresponds to precisely such a non-linear di eomorphism. Next, using the same prescription as for bulk-boundary propagators, we can lift this non-zero components of the 4D branch eld HIJ in Milne temporal gauge. The x 2 = e2 warp factor is the same one required in the bulk-boundary propagator for the 2D stress tensor. Since HIJ is a 4D di eomorphism around at space, we nd (x))dx dx = e2 ( d 2 + g~ij (y)dyidyj ): In conclusion, at the fully non-linear level, multiple subleading soft emissions are described by a branch H that encodes large di eomorphisms of the AdS3 metric. Since these soft perturbations of the metric are Milne zero modes, they couple to hard particle tracks according to d e6 T ij ( ; y) = where in the last line we have de ned Teij (y) = the Milne time-integrated stress tensor in a warped version of eq. (4.3). Equivalence to AdS3 gravity Similar to the case of gauge theory, we have seen that 4D soft graviton modes correspond to solutions of 3D gravity which are pure di eomorphisms. It is then expected that the resulting theory is topological, which is reasonable because gravity in AdS3 is famously equivalent to a CS theory, at least perturbatively [69]. In particular, one can de ne a non-abelian CS gauge eld, A a = ieia, where e is the dreibein, generators J a iKa, where J a and Ka are rotations and boosts, respectively. group of the CS theory is SL(2; C), corresponding to the global isometries of AdS3, or Via the embedding formalism, A a is associated with (+) and ( ) helicity soft gravigroup factorizes, so there is no intrinsic reason why we must restrict to a single helicity like we did for non-abelian CS theory. At the level of the dual CFT2 we are then permitted to compute mixed correlators involving both the holomorphic and anti-holomorphic stress tensor, t(z) and t(z). With the non-abelian structure clari ed, we can Laurent expand the holomorphic stress tensor into the in nite set of non-abelian Virasoro charges. Relatedly, the CS structure of the subleading soft amplitudes again implies that the dynamics of soft gravitons is governed by a non-abelian analog of the AB e ect, where the CS graviton eld is the now the eld encoding memory e ects. Unlike for electromagnetic memories, we have not as yet matched this kind of AB e ect in detail with the \spin memory" e ects discussed already in the nal note on the rigor of our conclusions here: what we have shown thus far is that LSZ reduction onto multiple subleading soft emissions. We have not yet proven that the scattering amplitudes of plane waves have the requisite commutativity amongst multiple subleading soft limits required for simultaneous LSZ reduction onto multiple bulk-boundary propagators. But we expect that the AdS3 gravity picture should identify any obstructions to multiple soft limits, as it did in non-abelian CS gauge theory for mixed soft helicities. While no such obstructions appear here, it would still be interesting to compute explicitly the commutativity properties of subleading graviton soft limits for these amplitudes in Minkowski space. Virasoro central charge from internal soft exchange The Virasoro central charge, c, is arguably the most important quantity in a 2D CFT [85]. In theories with semi-classical AdS3 duals, c is given by the AdS3 Planck scale in units of the AdS3 length. However, much like the gauge coupling in YM theory, the Planck scale enters simply as an overall factor in the gravity action, so it drops out of the homogeneous Einstein's equations. So at tree level, the soft branches characterizing multiple graviton emission are insensitive to the Planck scale and thus c. To make sense of c, we must then consider the gravitational analog of gure 7, which depicts a set of two hard processes exchanging a soft internal graviton. We interpret one process as a \measurement apparatus" for the subleading soft graviton emission of the CS level to the gauge coupling, here there is a dimensional mismatch between c and m2Pl. This means that an infrared length scale LIR does not decouple from the process. One can think of LIR as a formal scale separating \hard" from \soft". We thereby conclude that the Virasoro central charge scales as d3y pg~ (R~ + 1); Just this type of infrared sensitivity is present in the spin-memory e ect described in [45]. We can see this more directly by writing the 4D Einstein-Hilbert action in eq. (4.1) in terms of the 3D metric g~ij characterizing a soft branch in AdS3, as shown in eq. (4.25). Since g~ij is related by a di eomorphism to the pure AdS3 background metric, the resulting action should just be proportional to 3D gravity with a cosmological constant. The simple dependence of the action straightforwardly factors, yielding SEH = G R = 2 Z where we have taken \unit" dimensionally reduced AdS3 radius of curvature, in keeping with the normalization of our other formulas, and where late relates to LIR by We leave a formal analysis of the Virasoro central charge for future work. Although we are not carefully treating the physics underlying late here, we can nevertheless estimate the central charge from this rough scaling, From super-rotations to super-translations in Mink4 Let us now discuss the relation between our results and the asymptotic symmetries of Mink4. While there is an expansive literature on this subject, we will be quite brief here. Long ago, BMS [25, 26] discovered the existence of an in nite-dimensional symmetry of at space corresponding to super-translations at null in nity. Physically, these super-translations are di eomorphisms of retarded time that depend on angles on the celestial sphere. More recently, [27] argued that the super-translation algebra can be further extended to include super-rotations encoding an underlying Virasoro algebra. From their analysis of large di eomorphisms, they proposed an extended BMS algebra [71], [Lm; Ln] = (m [Pmn; Prs] = 0 [Lm; Prs] = dropping for the moment the Virasoro central charge. Here the Virasoro generators Lm correspond to the super-rotations while the generators Pmn correspond to super-translations. The Poincare sub-algebra is P00; P01; P10; P11; where the four super-translation generators are nothing more than the four components of later showed that the super-translations and super-rotations, at least at the level of single soft emission, arise from the leading and subleading Weinberg soft theorems. Here we will use eq. (4.33) as a guide for constructing super-translations as a combination of super-rotations and ordinary translations. While ordinary translations are quite obscure in Milne and Rindler coordinates, they are of course still a symmetry of at space, so they should also be global symmetries of the CFT. Since the 2D stress tensor is comprised of super-rotation generators, we can commute it with regular translations to obtain [t(z); P00] = m= 1 zm+2 [Lm; P00] = m= 1 zm+2 Pm0 = In analogy with the 2D CFT for gauge theory, we have de ned a super-translation current, j(z) = m= 1 which is holomorphically conserved, so @zj(z) = 0. We can use this result to determine the Ward identity for j(z). From our formula for the 2D stress tensor Ward identity in eq. (4.10), we already see an explicit connection to super-rotations through the angular momentum operators Ji Now taking the commutator of eq. (4.10) with P00, we obtain acting on the hard legs. h[t(z); P00]O(z1; z1) O(zn; zn)i = @z Comparing with eq. (4.35), we see that the Ward identity for the super-translation current is hj(z)O(z1; z1) Hence, we deduce that the charge associated with the super-translation Ward identity is the physical momentum in the q direction. Chern-Simons theory for super-translations? We have shown how 4D super-translations can be obtained from the 2D stress tensor t(z) via the commutation relations of the extended BMS algebra. Furthermore, we saw that correlators of t(z) correspond to a particular angular convolution of the subleading graviton soft theorem. Given the underlying connection of j(z) to super-translations, it is then quite natural for j(z) to relate to the leading graviton soft theorem. As we will see, this is indeed the case. To understand why, we revisit the auxiliary tensor primary t~(z; z) de ned in eq. (4.16), whose correlators are literally equal to the 4D subleading soft graviton factor. In particular, let us consider the CFT2 operator, [t~(z; z); P00], de ned by the commutator of this auxiliary tensor and regular translations. It is simple to see that the bulk-boundary propagator associated with the operator propagator for [t~(z; z); P00] is by de nition just the derivative of the bulk-boundary propagator of t~(z; z) in the q direction. Concretely, this implies that the bulk-boundary propagator for [t~(z; z); P00] is simply q @ K~ is the bulk-boundary propagator for t~(z; z). Since the latter is a pure di eomorphism, so too is the former. As we will see, this happens for a reason: this commutator is directly related to the holomorphic current for super-translations, j(z). covariant form. We can evaluate this expression using the fact that global translations acting on the hard legs, [Ji ; ki ] = ki Applying these relations, eq. (4.39) simpli es to Using our now standard methodology, let us compute the correlator for this commui=1 kki h[t~(z; z); P00]O(z1; z1) O(zn; zn)i; (4.41) again using that P00 = qP and P = Pin=1 ki . The above correlator simpli es to where qk = particle in the q direction. As advertised, the right-hand side of this expression as precisely the leading Weinberg soft graviton factor [33] in our variables. Comparing with eq. (4.40), we deduce that the holomorphic super-translation current is j(z) = @z[t~(z; z); P00]: Since the bulk-boundary propagator for [t~(z; z); P00] is a pure di eomorphism, so too is the one for j(z). This suggests that there should again be a \bulk" topological description of the holomorphic 2D super-translation current, sensitive to the passage of hard particles. While this result is encouraging, there are several reasons why such a topological description of super-translations cannot be a straightforward CS theory. First of all, from eq. (4.43), we see that j(z) is not a primary operator, as would be the case for the dual of a CS gauge eld, and is instead descendant from a commutator of t~(z; z). Relatedly, the global subgroup of super-translations, i.e. ordinary translations, transform under the SL(2; C) Lorentz group, unlike the global subgroup of a Kac-Moody algebra dual to a CS theory, which is SL(2; C) invariant. In any case, it would be very interesting to determine a bulk topological description for super-translations, if indeed one exists. Future directions A central result of this work is a recasting of 4D scattering amplitudes and their soft limits as correlators of a 2D CFT. In particular, we showed that soft elds in 4D gauge theory and gravity have a description in terms of 3D CS theory en route to a mapping onto aspects of 4D | soft theorems, asymptotic symmetries, and memory e ects | are elegantly encoded as 2D Ward identities, their associated Kac-Moody and Virasoro symmetries, and 3D Aharonov-Bohm type e ects. Of course, the results presented here are but a rst step theories for describing soft gauge and gravitational phenomena. Many questions remain, o ering numerous avenues for future work that we now discuss. First and foremost, we would like to better understand the role of unitarity in the 2D CFT, which cannot itself be unitary nor even a Wick rotation of a unitary CFT. Rather, since time is emergent, so too must be unitarity, which will then be non-manifest in the 2D description. On the other hand, starting from unitary 4D scattering amplitudes the 2D correlators must still somehow encode unitarity. However, what we really seek is some independent principle within the CFT guaranteeing 4D unitarity. Another open question relates to the role of 4D massive particles. The foliation approach taken here is in principle consistent with such a generalization, but there will surely be new subtleties. Certainly with massive particles, the Weyl invariance used to simplify even the free particle analysis will be lost, and a more general complex set of scaling dimensions will arise. Relatedly, massive particles will not actually reach null in nity, but must \sensed" sub-asymptotically. More involved will be an extension of our results to loop level, where our foliation approach should apply. With loops, it is likely that the CS description for soft gauge boson modes will have a level which depends on the infrared scale separating \hard" from \soft", due to the running of the gauge coupling. An obvious exception is if the gauge coupling is at an infrared xed point, in which case there may be a non-perturbative level free of infrared scale dependence. It would be interesting to understand whether the usual level quantization of CS theory implies that only 4D gauge theories with suitably quantized gauge couplings have a non-perturbative CS soft limit. Furthermore, it was shown in [86] and [87] that the subleading soft theorems of gauge theory and gravity are valid at tree level but are corrected at one-loop and higher. Interestingly, these corrections appear to be critically tied to infrared divergences [88]. This is naively quite disturbing because we saw that the subleading soft theorem for gravity is at the root of the Ward identities for the 2D stress tensor.7 However, more carefully examined, there need be no actual con ict. The Ward identity for the 2D stress tensor is related but not equal to the subleading graviton soft theorem, which is corrected at one loop. In fact, the complicated angular convolution in eq. (4.20) implies a highly non-trivial prescription for LSZ reduction that must be applied to the amplitude from the start. It is possible that at loop level, the 2D stress tensor continues to exist with some modi ed relationship to the Minkowski soft limit. In any case, it is of utmost importance to study the robustness of our picture at loop level. A distinct but related question is to what extent the subleading soft theorems for gauge theory and the subsubleading soft theorems for gravity | which are known to be universal at tree level | might arise within the structure of the CFT2. For example, from the CFT perspective, new non-conserved vector currents should robustly arise from taking the conserved limit of non-conserved tensor operators [89], which are AdS/CFT dual to the KK \graviphoton" of the e ective compacti cation implied by the soft limit. The OPE is a central feature of any CFT, which in the present context corresponds to the structure of 4D collinear singularities. This suggests that the CFT structure may facilitate some constructive method for building scattering amplitudes from collinear data. This is reminiscent of the BCFW recursion relations, which when reduced down to threeparticle amplitudes e ectively does this. On the other hand, the importance of self-dual 7The Ward identities for holomorphic conserved currents on the other hand arise from the leading gauge and gravity soft theorems which are not loop-corrected, and are therefore unthreatened. con gurations and the appearance of natural reference spinors discussion might naturally connect with CSW rule constructions for scattering amplitudes. The focus on soft limits and collinear singularities also suggests connections with softcollinear e ective theory [90, 91], which may well be important for a loop-level formulation of asymptotic symmetries and the ideas presented in this paper. There is also the question of whether our results can shed new light on the information paradox. As proposed in [67], soft \hair" could o er an intriguing caveat to the usual picture of black hole information loss. Nevertheless, stated purely in terms of soft radiation and gauge and gravitational memories, it is unclear how such a classical e ect can resolve the paradox. On the other hand, our results connect these e ects to Aharonov-Bohm e ects on the celestial sphere, which may o er a more quantum mechanical approach to this problem. Also deserving of further study is our toy model for black hole horizons coming from the Rindler horizon of Minkowski spacetime. In our picture, the restriction to the Rindler region revealed an extension of the CFT structure onto the past and future boundary of the horizon | e ectively the dS/CFT dual of the past and future wavefunction of the horizon. Here, the CFT gives a description of this horizon, extending the notion of asymptotic symmetries in its presence. It would be interesting if these features, especially those related to topological structure of memories, extended to real black holes in less symmetric spacetimes. Finally, it would be worthwhile to see if the foliation approach followed here can be applied to spacetimes other than Mink4, for example AdS4, to uncover new symmetries and topological features emerging in special limits. C.C. is supported by a Sloan Research Fellowship and a DOE Early Career Award under Grant No. DE-SC0010255. A.D. and R.S. are supported in part by the NSF under Grant No. PHY-1315155 and by the Maryland Center for Fundamental Physics. R.S. would also like to thank the Gordon and Betty Moore Foundation for the award of a Moore Distinguished Scholar Fellowship to visit Caltech, as well as the hospitality of the Walter Burke Institute for Theoretical Physics, where a substantial part of this work was completed. The authors are grateful to Nima Arkani-Hamed, Ricardo Caldeira Costa, Liam Fitzpatrick, Ted Jacobson, Dan Kapec, Jared Kaplan, Juan Maldacena, Ira Rothstein, and Anthony Speranza for useful discussions and comments. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE]. [hep-th/9802150] [INSPIRE]. string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE]. 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Clifford Cheung, Anton de la Fuente, Raman Sundrum. 4D scattering amplitudes and asymptotic symmetries from 2D CFT, Journal of High Energy Physics, 2017, 112, DOI: 10.1007/JHEP01(2017)112