#### 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].
Study Institute in Elementary Particle Physics (TASI 2010), (2010), pg. 3
[arXiv:1010.6134] [INSPIRE].
[6] R. Sundrum, From
xed points to the fth dimension, Phys. Rev. D 86 (2012) 085025
[arXiv:1106.4501] [INSPIRE].
(2011) 025 [arXiv:1011.1485] [INSPIRE].
127 [arXiv:1111.6972] [INSPIRE].
[arXiv:1112.4845] [INSPIRE].
QFT in AdS, arXiv:1607.06109 [INSPIRE].
Phys. B 665 (2003) 545 [hep-th/0303006] [INSPIRE].
(2005) 123 [hep-th/0405252] [INSPIRE].
Phys., Springer, New York U.S.A. (1997) [INSPIRE].
[arXiv:1308.0589] [INSPIRE].
JHEP 10 (2014) 112 [arXiv:1407.3789] [INSPIRE].
II: two dimensional amplitudes, arXiv:1607.06110 [INSPIRE].
21 [INSPIRE].
031602 [arXiv:1509.00543] [INSPIRE].
[arXiv:1312.2229] [INSPIRE].
graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].
quantum gravity S-matrix, JHEP 08 (2014) 058 [arXiv:1406.3312] [INSPIRE].
QED, Phys. Rev. Lett. 113 (2014) 111601 [arXiv:1407.3814] [INSPIRE].
symmetry in supersymmetric gauge theories, arXiv:1511.07429 [INSPIRE].
Rev. 110 (1958) 974 [INSPIRE].
(1968) 86 [INSPIRE].
[arXiv:1103.2981] [INSPIRE].
stars, Sov. Astron. 18 (1974) 17.
prospects, Nature 327 (1987) 123.
Rev. Lett. 67 (1991) 1486 [INSPIRE].
Quant. Grav. 30 (2013) 195009 [arXiv:1307.5098] [INSPIRE].
053 [arXiv:1502.06120] [INSPIRE].
Phys. Rev. 115 (1959) 485 [INSPIRE].
360 (1991) 362 [INSPIRE].
(1991) 802 [INSPIRE].
theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].
charges in AdS, Phys. Rev. D 85 (2012) 045010 [arXiv:1109.6010] [INSPIRE].
theorem, JHEP 07 (2015) 115 [arXiv:1505.05346] [INSPIRE].
spacetimes, JHEP 11 (2012) 046 [arXiv:1206.3142] [INSPIRE].
Mumbai India January 5{10 2001 [hep-th/0106109] [INSPIRE].
models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
JHEP 03 (2010) 133 [arXiv:0911.0043] [INSPIRE].
Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].
11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
[arXiv:1404.5625] [INSPIRE].
quantization of the Chern-Simons-Witten theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].
(1988) 46 [INSPIRE].
AdS3/CFT2 correspondence, hep-th/0403225 [INSPIRE].
[84] S. Oxburgh and C.D. White, BCJ duality and the double copy in the soft limit, JHEP 02
[72] D. Kapec, P. Mitra, A.-M. Raclariu and A. Strominger, A 2D stress tensor for 4D gravity,
elds in curved space, Camb. Monogr. Math.
Phys., Cambridge Univ. Press, Cambridge U.K. (1984) [INSPIRE].
gauged extended supergravity, Phys. Lett. B 115 (1982) 197 [INSPIRE].
[77] R. Monteiro and D. O'Connell, The kinematic algebra from the self-dual sector, JHEP 07
[78] Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys.
[79] S. Axelrod and I.M. Singer, Chern-Simons perturbation theory, in Di erential geometric
methods in theoretical physics. Proceedings, 20th International Conference, New York U.S.A.
[81] H. Leutwyler, A (2 + 1)-dimensional model for the quantum theory of gravity, Nuovo Cim. A
https://courses.edx.org/c4x/MITx/8.EFTx/asset/notes scetnotes.pdf.
[1] J.M. Maldacena , The large-N limit of superconformal eld theories and supergravity, Int . J.
[2] S.S. Gubser , I.R. Klebanov and A.M. Polyakov , Gauge theory correlators from noncritical string theory, Phys . Lett . B 428 ( 1998 ) 105 [hep-th /9802109] [INSPIRE].
[3] E. Witten , Anti-de Sitter space and holography, Adv. Theor. Math. Phys . 2 ( 1998 ) 253 [4] O. Aharony , S.S. Gubser , J.M. Maldacena , H. Ooguri and Y. Oz , Large-N [5] J. Polchinski , Introduction to gauge/gravity duality, in Proceedings, Theoretical Advanced [7] J. Penedones , TASI lectures on AdS/CFT, arXiv:1608 .04948 [INSPIRE].
[8] J. Polchinski , S matrices from AdS space-time , hep-th/ 9901076 [INSPIRE].
[9] L. Susskind , Holography in the at space limit , hep-th/ 9901079 [INSPIRE].
[10] M. Gary , S.B. Giddings and J. Penedones , Local bulk S-matrix elements and CFT singularities, Phys. Rev. D 80 ( 2009 ) 085005 [arXiv:0903.4437] [INSPIRE].
[11] J. Penedones , Writing CFT correlation functions as AdS scattering amplitudes , JHEP 03 [12] A.L. Fitzpatrick and J. Kaplan , Scattering states in AdS/CFT , arXiv:1104.2597 [INSPIRE].
[13] A.L. Fitzpatrick and J. Kaplan , Analyticity and the holographic S-matrix , JHEP 10 ( 2012 ) [14] A.L. Fitzpatrick and J. Kaplan , Unitarity and the holographic S-matrix , JHEP 10 ( 2012 ) 032 [15] M.F. Paulos , J. Penedones , J. Toledo , B.C. van Rees and P. Vieira , The S-matrix bootstrap I: [16] M.F. Paulos , J. Penedones , J. Toledo , B.C. van Rees and P. Vieira , The S-matrix bootstrap [17] J. de Boer and S.N. Solodukhin , A holographic reduction of Minkowski space-time , Nucl.
[18] S.N. Solodukhin , Reconstructing Minkowski space-time , IRMA Lect. Math. Theor. Phys . 8 [19] P.D. Francesco , P. Mathieu and D. Senechal , Conformal eld theory, Grad . Texts Contemp.
[20] A. Strominger , Asymptotic symmetries of Yang-Mills theory , JHEP 07 ( 2014 ) 151 [21] T. He , P. Mitra , A.P. Porfyriadis and A. Strominger , New symmetries of massless QED, [22] T. He , P. Mitra and A. Strominger , 2D Kac-Moody symmetry of 4D Yang-Mills theory , [23] D. Kapec , M. Pate and A. Strominger , New symmetries of QED, arXiv:1506.02906 [24] A. Strominger , Magnetic corrections to the soft photon theorem , Phys. Rev. Lett . 116 ( 2016 ) [25] H. Bondi , M.G.J. van der Burg and A.W.K. Metzner , Gravitational waves in general relativity . 7. Waves from axisymmetric isolated systems , Proc. Roy. Soc. Lond. A 269 ( 1962 ) [26] R.K. Sachs , Gravitational waves in general relativity . 8. Waves in asymptotically at space-times , Proc. Roy. Soc. Lond. A 270 ( 1962 ) 103 [INSPIRE].
[27] G. Barnich and C. Troessaert , Symmetries of asymptotically at 4 dimensional spacetimes at null in nity revisited , Phys. Rev. Lett . 105 ( 2010 ) 111103 [arXiv:0909.2617] [INSPIRE].
[28] A. Strominger , On BMS invariance of gravitational scattering , JHEP 07 ( 2014 ) 152 [29] T. He , V. Lysov , P. Mitra and A. Strominger , BMS supertranslations and Weinberg's soft [30] D. Kapec , V. Lysov , S. Pasterski and A. Strominger , Semiclassical Virasoro symmetry of the [31] V. Lysov , S. Pasterski and A. Strominger , Low's subleading soft theorem as a symmetry of [32] T.T. Dumitrescu , T. He , P. Mitra and A. Strominger , In nite-dimensional fermionic [33] S. Weinberg , Infrared photons and gravitons , Phys. Rev . 140 ( 1965 ) B516 [INSPIRE].
[34] F.E. Low , Bremsstrahlung of very low-energy quanta in elementary particle collisions , Phys.
[35] T.H. Burnett and N.M. Kroll , Extension of the low soft photon theorem , Phys. Rev. Lett. 20 [36] C.D. White, Factorization properties of soft graviton amplitudes , JHEP 05 ( 2011 ) 060 [37] F. Cachazo and A. Strominger , Evidence for a new soft graviton theorem , arXiv:1404. 4091 [38] L. Bieri and D. Gar nkle, An electromagnetic analogue of gravitational wave memory , Class.
[39] S. Pasterski , Asymptotic symmetries and electromagnetic memory , arXiv:1505. 00716 [40] L. Susskind , Electromagnetic memory, arXiv:1507 .02584 [INSPIRE].
[41] Y. Zeldovich and A. Polnarev , Radiation of gravitational waves by a cluster of superdense [42] V.B. Braginsky and K.S. Thorne , Gravitational-wave bursts with memory and experimental [43] D. Christodoulou , Nonlinear nature of gravitation and gravitational wave experiments , Phys.
[45] S. Pasterski , A. Strominger and A. Zhiboedov , New gravitational memories, JHEP 12 ( 2016 ) [46] Y. Aharonov and D. Bohm , Signi cance of electromagnetic potentials in the quantum theory , [47] G.W. Moore and N. Read , Nonabelions in the fractional quantum Hall e ect, Nucl . Phys . B [48] X.G. Wen , Non-Abelian statistics in the fractional quantum Hall states , Phys. Rev. Lett. 66 [49] Y.-T. Chien , M.D. Schwartz , D. Simmons-Du n and I.W. Stewart , Jet physics from static [50] M. Campiglia and A. Laddha , Asymptotic symmetries of QED and Weinberg 's soft photon [51] R.N.C. Costa , Holographic reconstruction and renormalization in asymptotically Ricci - at [52] A. Strominger , The dS/CFT correspondence, JHEP 10 ( 2001 ) 034 [hep-th /0106113] [53] E. Witten , Quantum gravity in de Sitter space , in Strings 2001 : International Conference, [54] J.M. Maldacena , Non-Gaussian features of primordial uctuations in single eld in ationary [55] D. Anninos , T. Hartman and A. Strominger , Higher spin realization of the dS /CFT correspondence, Class. Quant. Grav. 34 ( 2017 ) 015009 [arXiv:1108.5735] [INSPIRE].
[56] P.A.M. Dirac , Wave equations in conformal space , Annals Math . 37 ( 1936 ) 429 [INSPIRE].
[57] L. Cornalba , M.S. Costa and J. Penedones , Deep inelastic scattering in conformal QCD , [58] S. Weinberg , Six-dimensional methods for four-dimensional conformal eld theories , Phys.
[59] M.S. Costa , J. Penedones , D. Poland and S. Rychkov , Spinning conformal correlators , JHEP [60] M.S. Costa , V. Goncalves and J. Penedones , Spinning AdS propagators, JHEP 09 ( 2014 ) 064 [61] V.P. Nair , A current algebra for some gauge theory amplitudes , Phys. Lett . B 214 ( 1988 ) 215 [62] E. Witten , Perturbative gauge theory as a string theory in twistor space, Commun . Math.
[63] E. Witten , Quantum eld theory and the Jones polynomial, Commun . Math. Phys. 121 [64] S. Elitzur , G.W. Moore , A. Schwimmer and N. Seiberg , Remarks on the canonical [65] E. Witten , On holomorphic factorization of WZW and coset models, Commun . Math. Phys.
[66] S. Gukov , E. Martinec , G.W. Moore and A. Strominger , Chern-Simons gauge theory and the [67] S.W. Hawking , M.J. Perry and A. Strominger , Soft hair on black holes , Phys. Rev. Lett. 116 [68] D. Harlow , Jerusalem lectures on black holes and quantum information, Rev. Mod. Phys . 88 [82] A. Staruszkiewicz , Gravitation theory in three-dimensional space , Acta Phys. Polon . 24 [83] H. Kawai , D.C. Lewellen and S.-H. Henry Tye , A relation between tree amplitudes of closed [70] E. Witten , Three-dimensional gravity revisited , arXiv:0706 .3359 [INSPIRE].
[71] G. Barnich and C. Troessaert , BMS charge algebra , JHEP 12 ( 2011 ) 105 [arXiv:1106.0213] [73] N.D. Birrell and P.C.W. Davies , Quantum [74] P. Breitenlohner and D.Z. Freedman , Positive energy in anti-de Sitter backgrounds and [75] P. Breitenlohner and D.Z. Freedman , Stability in gauged extended supergravity , Annals Phys.
[76] P.K. Townsend , K. Pilch and P. van Nieuwenhuizen, Selfduality in odd dimensions , Phys.
[80] S. Axelrod and I.M. Singer , Chern-Simons perturbation theory. II, J. Di . Geom . 39 ( 1994 ) [69] E. Witten , ( 2 + 1 ) -dimensional gravity as an exactly soluble system , Nucl. Phys . B 311 [85] J. Cardy , The ubiquitous `c': from the Stefan-Boltzmann law to quantum information , J.
Stat. Mech. 10 ( 2010 ) P10004 [arXiv:1008 .2331] [INSPIRE].
[86] Z. Bern , S. Davies and J. Nohle , On loop corrections to subleading soft behavior of gluons and gravitons , Phys. Rev. D 90 ( 2014 ) 085015 [arXiv:1405.1015] [INSPIRE].
[87] S. He , Y.-T. Huang and C. Wen , Loop corrections to soft theorems in gauge theories and gravity , JHEP 12 ( 2014 ) 115 [arXiv:1405.1410] [INSPIRE].
[88] Z. Bern , S. Davies , P. Di Vecchia and J. Nohle , Low-energy behavior of gluons and gravitons from gauge invariance , Phys. Rev. D 90 ( 2014 ) 084035 [arXiv:1406.6987] [INSPIRE].
[89] J. Maldacena and A. Zhiboedov , Constraining conformal eld theories with a slightly broken higher spin symmetry , Class. Quant. Grav . 30 ( 2013 ) 104003 [arXiv:1204.3882] [INSPIRE].
[90] I. Stewart and C. Bauer , The soft-collinear e ective theory , [91] A.J. Larkoski , D. Neill and I.W. Stewart , Soft theorems from e ective eld theory , JHEP 06