A ddimensional stress tensor for Minkd+2 gravity
Accepted: May
Minkd+2 gravity
Daniel Kapec 0 2 3
Prahar Mitra 0 1 2 3
Quantum Gravity, Scattering Amplitudes
0 Center for the Fundamental Laws of Nature, Harvard University
1 School of Natural Sciences, Institute for Advanced Study
2 Cambridge , MA, 02138 U.S.A
3 Princeton, NJ , 08540 U.S.A
We consider the treelevel scattering of massless particles in (d+2)dimensional asymptotically at spacetimes. The Smatrix elements are recast as correlation functions of local operators living on a spacelike cut Md of the null momentum cone. The Lorentz group SO(d + 1; 1) is nonlinearly realized as the Euclidean conformal group on Md. Operators of nontrivial spin arise from massless particles transforming in nontrivial representations of the little group SO(d), and distinguished operators arise from the softinsertions of gauge bosons and gravitons. The leading softphoton operator is the shadow transform of a conserved spinone primary operator Ja, and the subleading softgraviton operator is the shadow transform of a conserved spintwo symmetric traceless primary operator Tab. The universal form of the softlimits ensures that Ja and Tab obey the Ward identities expected of a conserved current and energy momentum tensor in a Euclidean CFTd, respectively.
Conformal Field Theory; Field Theories in Higher Dimensions; Models of

A
1 Introduction 2 3 4
Lorentz transformations and the conformal group
Conserved currents and soft theorems
4.1
4.2
4.3
Leading softphoton theorem and the conserved U(1) current
Subleading softgraviton theorem and the stress tensor
Leading softgraviton theorem and momentum conservation
5
Conclusion
A Spacetime picture
1
Introduction
[ I , is a null cone
with a (possibly singular) vertex at spatial in nity. Massless excitations propagating in
such a spacetime pass through I at isolated points on the celestial sphere. Guided by
the holographic principle, one might hope that the Smatrix for the scattering of massless
particles in asymptotically at spacetimes in (d + 2)dimensions might be reexpressed as a
collection of correlation functions of local operators on the celestial sphere Sd at null in nity,
with operator insertions at the points where the particles enter or exit the spacetime. The
Lorentz group would then be realized as the group of conformal motions of the celestial
sphere, and the Lorentz covariance of the Smatrix would guarantee that the local operators
have wellde ned transformation laws under the action of the Euclidean conformal group
SO(d + 1; 1). On these general grounds one expects the massless Smatrix to display some
of the features of a ddimensional Euclidean conformal eld theory (CFTd).
+
It has recently become possible to make some of these statements more precise in four
dimensions, due in large part to Strominger's infrared triangle that relates soft theorems,
asymptotic symmetry groups and memory e ects [1{40]. While the speci c details of a
putative holographic formulation are expected to be model dependent, it should be possible
to make robust statements (primarily regarding symmetries) based on universal properties
of the Smatrix. One interesting class of universal statements about the Smatrix concerns
the socalled softlimits [41{48] of scattering amplitudes. In the limit when the wavelength
of an external gauge boson or graviton becomes much larger than any scale in the scattering
process, the Smatrix factorizes into a universal soft operator (controlled by the soft particle
{ 1 {
and the quantum numbers of the hard particles) acting on the amplitude without the soft
insertion. This sort of factorization is reminiscent of a Ward identity, and indeed in four
dimensions the softphoton, softgluon, and softgraviton theorems have been recast in the
form of Ward identities for conserved operators in a putative CFT2 [2, 16, 28, 31, 34, 49,
50]. Most importantly for the present work, in [28, 31, 50] an operator was constructed
from the subleading softgraviton theorem whose insertion into the four dimensional
Smatrix reproduces the Virasoro Ward identities of a CFT2 energy momentum tensor. The
subleading softgraviton theorem holds in all dimensions [51{57], so it should be possible
to construct an analogous operator in any dimension. We will see that this is indeed the
case, and that the construction is essentially xed by Lorentz (conformal) invariance.
massless particle kinematics and describe the map from the (d + 2)dimensional Smatrix
to a set of ddimensional \celestial correlators" de ned on a spacelike cut of the null
momentum cone. Section 3 describes the realization of the Euclidean conformal group on
these correlation functions in terms of the embedding space formalism. Section 4 outlines
the construction of conserved currents  namely the conserved U(1) current and the
stress tensor  in the boundary theory and their relations to the leading softphoton and
subleading softgraviton theorems. Section 5 concludes with a series of open questions. In
appendix A, we brie y discuss the bulk spacetime interpretation of our results and their
relations to previous work.
2
Massless particle kinematics
The basic observable in asymptotically at quantum gravity is the Smatrix element
S = h out j in i
(2.1)
(2.2)
(2.3)
between an incoming state on past null in nity (
I ) and an outgoing state on future
null in nity (I+). The perturbative scattering states in asymptotically at spacetimes are
characterized by collections of well separated, noninteracting particles.1 Each massless
particle is characterized by a null momentum p and a representation of the little group
SO(d), as well as a collection of other quantum numbers such as charge, avor, etc. Null
space Rd+1;1,
momenta are constrained to lie on the future light cone C
+ of the origin in momentum
C
+ = fp 2 Rd+1;1 p2 = 0 ; p0 > 0g :
A convenient parametrization for the momentum, familiar from the embedding space
formalism in conformal eld theories [58, 59], is given by
p (!; x) = ! (x)p^ (x) ;
p^ (x) =
1In four dimensions, the probability to scatter into a state with a
nite number of gauge bosons or
where x2 = xaxa = abxaxb. The metric on this null cone is degenerate and is given by
(x) de nes the conformal factor on Md. In most of what follows we will choose (x) = 1
for computational simplicity, although the generalization to an arbitrary conformally at
Euclidean cut is straightforward.2 The (d+2)dimensional Lorentzinvariant measure takes
while the Lorentzian inner product is given by
Massless particles of spin s can be described by symmetric traceless elds
"a11::::::ass (p) Oa1:::as (p)eip X + Oay1:::as (p)e ip X i
h
satisfying the equations
Under gauge transformations,
We will work in the gauge
X
1::: s (X) = 0 ;
n 1
1::: s (X) = 0 ;
n = (1; 0a; 1) :
A natural basis for the vector representation of the little group SO(d) is given in terms of
the d polarization vectors
"a (x)
These are orthogonal to both n and p^ and satisfy
"a(x) "b(x) = ab ;
"a (x)"a(x) =
(x)
+ n p^ (x) + n p^ (x) :
2Other choices of (x) have also proved useful in previous analyses. In particular, the authors of [3, 4,
8, 18] choose
(x) = 2(1 + x2) 1, yielding the round metric on Sd. For nonconstant
(x), ddimensional
by their conformally covariant counterparts, the GJMS operators [60].
1 h"a (x)"b (x) + "b (x)"a(x)
i
1 ab
d
(x) :
The Fock space of massless scattering states is generated by the algebra of single
the energy direction actually appears spacelike rather than timelike.3 The creation and
annihilation operators can be viewed as operators inserted at speci c points of Md carrying
an additional quantum number !, so that the Smatrix takes the form of a conformal
correlator of primary operators. In other words, the amplitude with m incoming and
n
m outgoing particles4
An;m = hpm+1; : : : ; pnjp1; : : : ; pmi
can be equivalently represented as a correlation function on Md
An = h O1(p1) : : : On(pn) iMd = h O1(!1; x1) : : : On(!n; xn) iMd :
In this representation, outgoing states have ! > 0 and ingoing states have ! < 0. In the
rest of this paper, we will freely interchange the notation pi , (!i; xi) to describe the
insertion of local operators.
In appendix A, we demonstrate that the spacelike cut Md of the momentum cone is
naturally identi ed with the cross sectional cuts of I+. We construct the bulk coordinates
whose limiting metric on future null in nity is that of a null cone with crosssectional
metric (2.5). This provides a holographic interpretation of our construction, recasting
scattering amplitudes in asymptotically at spacetimes in terms of a \boundary" theory
that lives on I+.
3This is also the case for the null direction on I in asymptotic quantization.
4Here, we have suppressed all other quantum numbers that label the oneparticle states, such as the
polarization vectors, avor indices, or charge quantum numbers.
{ 4 {
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
In this section, we make explicit the map from (d + 2)dimensional momentum space
Smatrix elements to conformal correlators on the Euclidean manifold Md. The setup is
mathematically similar to the embedding space formalism, although in this case the
(d+2)dimensional \embedding space" is the physical momentum space rather than merely an
auxiliary ambient construction. The Lorentz group SO(d + 1; 1) with generators M
acts
linearly on momentum space vectors p
group of Md is given by the identi cations
2 Rd+1;1. The isomorphism with the conformal
Jab = Mab ;
Ta = M0;a
Md+1;a ;
D = Md+1;0 ;
Ka = M0;a + Md+1;a : (3.1)
The Jab generate SO(d) rotations, D is the dilation operator, and Ta and Ka are the
generators of translations and special conformal transformations, respectively. These operators
satisfy the familiar conformal algebra
[Jab; Jcd] = i( acJbd + bdJac
bcJad
adJbc) ;
HJEP05(218)6
Equation (3.1) describes the precise map between the linear action of Lorentz
transformations on p and nonlinear conformal transformations on (!; x). The latter amount to
transformations x ! x0(x) for which
Using (3.1), the conformal properties of the operators O(!; x) (we suppress spin labels
for convenience) follow from their Lorentz transformations,
[O(p); M
] = L
O(p) + S
O(p) ;
[O(p); P ] =
p O(p) ;
denotes the spins representation of the Lorentz group. We would like to rewrite
these relations in a way that manifests the action on Md. First, we note that
2xa ;
2xaxb
2
along with
where
and S
[Jab; Tc] = i( acTb
[Jab; Kc] = i( acKb
[D; Ta] =
iTa ;
[D; Ka] = iKa ;
bcTa) ;
bcKa) ;
[Ta; Kb] =
2i( abD + Jab) :
! ! !0 =
!
(x)
:
L
=
i p
(3.5)
(3.6)
2xa
La;d+1 =
i
2
i
2
(1
Lab =
(3.9)
HJEP05(218)6
(3.8)
(3.10)
(3.11)
(3.12)
The action of the spin matrix S
can be conveniently expressed in terms of the polarization
vectors (2.12). For instance, the action on a spin1 state is given by
In general one nds [
where Sab is the representation of the massless little group SO(d). The action of the
conformal group on the creation and annihilation operators is then given by
[O(!; x); Jab] =
We recognize these commutation relations as the de ning properties of a spins conformal
primary operator, with a nonstandard dilation eigenvalue
=
the momentum cone along its null direction.5
The fact that
is realized as a derivative simply re ects the fact that the energy eigenstates
do not diagonalize the dilation operator, which simply translates the spacelike cut Md of
4
Conserved currents and soft theorems
The operator content and correlation functions of the theory living on Md are highly
dependent on the spectrum and interactions of the (d + 2)dimensional theory under
consideration. However, the universal properties of the (d + 2)dimensional Smatrix are expected
to translate into general, model independent features of the \boundary theory". In this
section, we explore the consequences of the universal soft factorization properties of
Smatrix elements. Soft factorization formulas closely resemble Ward identities, and indeed
many soft theorems are known to be intimately related to symmetries of the Smatrix. The
existence of the soft theorems should therefore enable one to construct associated conserved
currents for the theory living on Md. The appropriate currents were constructed for d = 2
in [2, 16, 28, 31, 34, 50]. Here, we generalize these results to d > 2.
RC d!!
1O(!; x) for some contour C in the complex ! plane.
5It is possible to obtain standard conformal primary operators via a Mellin transform, O( ; x) =
{ 6 {
The leading softphoton theorem is a universal statement about the behavior of Smatrix
elements in the limit that an external photon's momentum tends to zero. It is model
independent, exists in any dimension, and states that
h Oa(q)O1(p1) : : : On(pn) i
n
From (4.3), we see that Sa(x) is a conformal primary operator with ( ; s) = (1; 1). Note
also that Sa(x) satis es
identically, without contact terms. In even dimensions,6 the leading softphoton theorem
is known [8, 12, 13, 17, 18] to be completely equivalent to the invariance of the Smatrix
under a group of angle dependent U(1) gauge transformations with noncompact support.
In d = 2, this symmetry is generated by the action of a holomorphic boundary current
Jz satisfying the appropriate KacMoody Ward identities (see [16]). In higher dimensions,
one consequently expects to encounter a conformal primary operator Ja(x) with ( ; s) =
(d
1; 1) satisfying the Ward identity
h @bJb(y)O1(!1; x1) : : : On(!n; xn) i = X Qk (d)(y
xk)h O1(!1; x1) : : : On(!n; xn) i :
n
k=1
n
k=1
{ 7 {
6From this point on, we will consider only the even dimensional case in order to avoid discussion of
fractional powers of the Laplacian.
operator"
the form
where Oa(!; x) creates an outgoing photon of momentum p(!; x) and polarization "a (x),
and Qk is the charge of the kth particle. We will rst de ne the \leading softphoton
HJEP05(218)6
Sa(x) = !li!m0 !Oa(!; x) :
Insertions of this operator are controlled by the leading softphoton theorem (4.1) and take
h Sa(x)O1(!1; x1) : : : On(!n; xn) i = @a X Qk log (x
xk)2 h O1(!1; x1) : : : On(!n; xn) i:
Our goal is to construct the conserved current Ja(x) from the soft operator Sa(x). The
inverse problem  constructing Sa(x) from an operator Ja(x) satisfying (4.5)  is easily
y)2], we nd
Z
xk)2]h O1(!1; x1) : : : On(!n; xn) i
= h Sa(x)O1(!1; x1) : : : On(!n; xn) i :
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
We therefore identify7
where Iab(x
y) is the conformally covariant tensor
Z
Sa(x) =
Z
ddy Iab(x
(x
y)y2) J b(y) ;
Iab(x
y) = ab
2
(x
y)a(x
(x
y)2
y)b
:
This nonlocal relationship between the
= 1 primary Sa and the
= d
1 primary Ja is
known as a shadow transform. For a spins operator of dimension
, the shadow operator
HJEP05(218)6
is given by [58]
Oea1:::as (x) = ab11::::::bass Z
ddy Ib1c1 (x
[(x
y)2]d
y) : : : Ibscs (x
y) Oc1:::cs (y) :
Here, ab11::::::bass is the invariant identity tensor in the spins representation,
ab11::::::bass = ffab11 ab22 : : : abssgg ;
where the notation f ; g denotes the symmetric traceless projection on the indicated indices.
The shadow transform is the unique integral transform that maps conformal primary
operators with ( ; s) onto conformal primary operators with (d
; s). Given that Sa has
( ; s) = (1; 1) while Ja has ( ; s) = (d
1; 1) it seems natural to expect the appearance of the shadow transform.
The shadow transform is, up to normalization [58, 61], its own inverse8
(4.7)
(4.8)
(4.9)
(4.10)
d
Using this, we can immediately write
(
d
(
1)(d
1 + s)(d
1)
d
2
1 + s) ( ) (d
Sa(x) = 2Jea(x) ;
1
Ja(x) =
2c(1; 1) Sea(x) :
Interestingly, the property (4.4) allows one to obtain a local relation between Ja(x)
and Sa(x)
Ja(x) =
1
(4 )d=2 (d=2)
(
) d2 1Sa(x) :
It is straightforward to verify that insertions of Ja(x) are given by
hJa(x)O1(!1; x1) : : : On(!n; xn)i =
2 d=2
(d=2) Xn Qk x
(x
k=1
j
xk)a
xkjd h O1(!1; x1) : : : On(!n; xn) i
and satisfy (4.5).
which do not a ect the Ward identity (4.5).
8The spatial integrals involved here are formally divergent and are regulated by the i prescription.
{ 8 {
In summary, we nd that the leading softphoton theorem in any dimension implies
the existence of a conserved current Ja(x) on the spatial cut Md. This current is
constructed as the shadow transform of the softphoton operator Sa(x). This correspondence
is reminiscent of a similar construction in AdS/CFT, where again the presence of a massless
bulk gauge eld produces a dual conserved boundary current. There have been attempts
to make this analogy more precise using a socalled \holographic reduction" of Minkowski
space [50, 62]. The (d + 2) dimensional Minkowski space can be foliated by hyperboloids
(and a null cone), each of which is invariant under the action of the Lorentz group. Inside
the light cone, this amounts to a foliation using a family of Euclidean AdSd+1, all sharing
an asymptotic boundary given by the celestial sphere.9 Performing a KaluzaKlein
reduction on the (timelike, noncompact) direction transverse to the AdSd+1 slices decomposes
the gauge eld A(X) in Minkowski space into a continuum of AdSd+1 gauge elds A!(x)
with masses
! (! is the socalled Milne energy). The ! ! 0 gauge eld  equivalent
to the soft limit  is massless in AdSd+1 and therefore induces a conserved current on the
ddimensional boundary. The holographic dictionary suggests that in the boundary theory,
one has a coupling of the form
Z
ddxSa(x)Ja(x) :
(4.15)
The discussion here suggests that a deeper holographic connection (beyond the
simple existence of conserved currents) may exist between the theory on Md and dynamics
in Minkowski space.
While intriguing, much remains to be done in order to elucidate this relationship. The hypothetical boundary theory is expected to have many peculiar properties, one of which we discuss in section 4.3.
4.2
Subleading softgraviton theorem and the stress tensor
In the previous section, we demonstrated that the presence of gauge elds in Minkowski
space controls the global symmetry structure of the putative theory on Md. As in AdS/
CFT, more interesting features arise when we couple the bulk theory to gravity and
consider gravitational perturbations. Flat space graviton scattering amplitudes also display
universal behavior in the infrared that is model independent and holds in any dimension.
Of particular interest here is the subleading softgraviton theorem, which states10
k=1
i Xn "ab pk q
pk q
!!0
Jk hO1(p1) : : : On(pn)i : (4.16)
Here Oab(q) creates a graviton with momentum q and polarization "ab (q), and Jk
is the
total angular momentum operator for the kth particle. The operator (1 + !@!) projects
out the Weinberg pole [45], yielding a nite ! ! 0 limit.
9This construction seems more natural in momentum space, where it is only the interior of the light
cone which is physically relevant, and one never needs to discuss the timelike de Sitter hyperboloids lying
outside the light cone.
10We work in units such that p8 G = 1.
{ 9 {
Returning to the analogy with AdSd+1=CFTd, one might expect that the bulk
softgraviton is associated to a boundary stress tensor, just as the bulk softphoton is related to
a boundary U(1) current. In a quantum
eld theory, the stress tensor generates the action
of spacetime (conformal) isometries on local operators. As we saw in (3.11), the angular
momentum operator J
generates these transformations on the local operators on Md.
Therefore, it is natural to suspect that the bulk subleading softgraviton operator
!!0
is related to the boundary stress tensor. Such a relationship was derived in four dimensions
Insertions of Bab(x) are controlled by the subleading softgraviton theorem (4.16) and
take the form (see (3.8) and (3.10) for the explicit forms of the orbital and spin angular
momentum operators)
hBab(x)O1(!1; x1) : : : On(!n; xn)i =
Pcab(x
From (4.18), we see that Bab(x) is a conformal primary operator with ( ; s) = (0; 2). One
can also check that
As in section 4.1, it is easiest to rst determine Bab in terms of Tab. Recall that the
Ward identities for the energy momentum tensor of a CFTd take the form [63]
h @dTdc(y)O1(!1; x1) : : : On(!n; xn) i =
(d)(y
xk)@xck h O1(!1; x1) : : : On(!n; xn) i ;
where
Pcab(x) =
1
2
xa bc + xb ac +
2 c
d
x ab
x2
4 xcxaxb :
h T cc(y)O1(!1; x1) : : : On(!n; xn) i =
(d)(y
xk)!k@!k h O1(!1; x1) : : : On(!n; xn) i ;
h T [cd](y)O1(!1; x1) : : : On(!n; xn) i =
(d)(y
xk)Skcdh O1(!1; x1) : : : On(!n; xn) i :
n
X
k=1
2
n
X
k=1
n
X
k=1
i Xn
2
k=1
1
xk)Skcd hO1(!1; x1) : : : On(!n; xn)i ;
(4.17)
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
Multiplying (4.21) by
y), and taking the sum, one nds
Tefabg(x) :
Bab(x) =
y)T cd(y)
ddyIfafc(x
y)Idgbg(x
y)T cd(y)
Once again, the soft operator appears as the shadow transform of a conserved current.
The relationship could have been guessed from the outset based on the dimensions of Bab
and Tfabg.11
Having derived (4.24), we can now invert the shadow transform to nd
Tfabg(x) =
1
c(0; 2) Beab(x) :
Z
ddxBab(x)Tab(x)
The shadow relationship between the soft operator Bab and the energy momentum
tensor is again suggestive of a coupling
in some hypothetical dual formulation of asymptotically
at gravity: the softgraviton
creates an in nitesimal change in the boundary metric, sourcing the operator Tab. In [28],
it was viewed as a puzzle that the energy momentum tensor appears nonlocal when written
in terms of the softmodes of the four dimensional gravitational eld. Here we see that this
is essentially the consequence of a linear response calculation, and that the nonlocality is
actually the only one allowed by conformal symmetry.
As in [28], it is possible to derive a local di erential equation for Tfabg in even
dimensions. We rst de ne the following derivative operator
DaOab
1
2(4 )d=2 (d=2 + 1)
One can check that
Then, acting on the rst equation of (4.24) with Da, we nd
d
d
(4.27)
DaPcab =
c (d)(x) :
b
11Note that only the symmetric traceless part of the stress tensor appears in this dictionary since the
graviton lies in the symmetric traceless representation of the little group. The trace term may be related
to softdilaton theorems.
(4.24)
(4.25)
(4.26)
(4.28)
(4.29)
In the previous two subsections, we have avoided the discussion of currents related to the
leading softgraviton theorem. This soft theorem is associated to spacetime translational
invariance (and more generally to BMS supertranslations [3, 4, 15, 20]). For scattering
amplitudes in the usual plane wave basis, this symmetry is naturally enforced by a
momentum conserving Dirac delta function. In our discussion above, we have chosen to make
manifest the Lorentz transformation properties of the scattering amplitude. Consequently,
translation invariance, or global momentum conservation, is unwieldy in our formalism. In
fact, it must somehow appear as a nonlocal constraint on the correlation functions on Md,
since arbitrary operator insertions corresponding to arbitrary con gurations of incoming
and outgoing momenta will in general violate momentum conservation. The di culty can
also be seen at the level of the symmetry algebra. Momentum conservation cannot arise
simply as a global Rd+1;1 symmetry of the CFTd, since the associated conserved charges
do not commute with the conformal (Lorentz) group. In light of this it is not clear that
our construction can really be viewed as a local conformal eld theory living on Md.12 We
have tried, unsuccessfully, to
nd a natural set of operators whose shadow reproduces the
leading softgraviton theorem13
!li!m0 !hOab(q)O1(p1) : : : On(pn)i = ! Xn "ab p p
k k
pk q hO1(p1) : : : On(pn)i :
k=1
The soft operator
has insertions given by
Gab(x) = lim !Oab(!; x)
!!0
h
@aUa(x)O1(!1; x1) : : : On(!n; xn)i =
xk)hO1(!1; x1) : : : On(!n; xn)i :
12However, it is also not clear that we should expect a local QFT dual to asymptotically at quantum
gravity. The
at space BekensteinHawking entropy is always superHagedorn in d
4. The high energy
density of states grows faster than in any local theory.
13It has also been suggested [64] that translational invariance of the Smatrix is realized through null
state relations of boundary correlators rather than through local current operators, since the former are
typically nonlocal constraints on CFT correlation functions.
k=1
d
2 Xn !kIa(db)(x
d xaxb
x2 :
1)
(
(d)(x
n
X !k
k=1
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
(4.35)
Ia(db)(x) = ab
1
(4 )d=2 (d=2) (d
hGab(x)O1(!1; x1) : : : On(!n; xn)i =
xk)hO1(!1; x1) : : : On(!n; xn)i ;
Thus, Ua(x) satis es the current Ward identity corresponding to \energy" conservation.
However, since ! is not a scalar charge, the current Ua(x) is not a primary operator (though
it has a wellde ned scaling dimension
= d). Acting on (4.35) with
2 R ddxIa(db)(y
d
x),
one nds
Gab(x) =
(4.36)
Unlike the U(1) current and the stress tensor, the leading softgraviton \current" is not
related to the soft operator Gab through a shadow transform. It may be possible to
interpret (4.36) as some other (conformally) natural nonlocal transform of Uc(x), but we do
not pursue this here.14
5
In this paper we have taken steps to recast the (d + 2)dimensional Smatrix as a collection
of celestial correlators, but many open questions remain. Our analysis relied on symmetry
together with the universal behavior of the Smatrix in certain kinematic regimes. It would
be interesting to analyze the consequences of other universal properties of the Smatrix
for the celestial correlators. The analytic structure and unitarity of the Smatrix should
be encoded in properties of these correlation functions, although the mechanism may be
subtle. It seems likely that the collinear factorization of the Smatrix could be used to
de ne some variant of the operator product expansion for local operators on the light cone.
Although this paper only addressed single soft insertions, double soft limits, appropriately
de ned, could be used to de ne OPE's between the conserved currents and stress tensors.
We expect supergravity soft theorems to yield a variety of interesting operators, including
a supercurrent. Finally, the interplay of momentum conservation with the CFTd structure
requires further clari cation. We leave these questions to future work.
Acknowledgments
We are grateful to Sabrina Pasterski, Abhishek Pathak, AnaMaria Raclariu, ShuHeng
Shao, Andrew Strominger, Xi Yin, and Sasha Zhiboedov for discussions. This work was
supported in part by DOE grant DESC0007870 and the Fundamental Laws Initiative at
Harvard. PM gratefully acknowledges support from DOE grant DESC0009988.
A
Spacetime picture
In this appendix we construct coordinates for Minkd+2 whose limiting metric on I
cross sectional cuts given by Md. Consider the coordinate transformation from the at
+ has
14The shadow transform ( ; s) ! (d
; s) is related to the Z2 symmetry of the quadratic and quartic
Casimirs of the conformal group c2 =
(d
) + s(2
d
s), c4 =
s(2
d
and c4 are also invariant under another Z2 symmetry under which ( ; s) ! (1
s)(
s; 1
1)(d
1). c2
). Equation (4.36)
may be the integral representation of a shadow transform followed by the second Z2 transform which maps
(d + 1; 0) ! ( 1; 0) ! (1; 2).
to the coordinates (u; r; xa) given by
where p (xa) is given by (2.3) with ! set to one. We have
X (u; r; xa) = rp (xa) + uk (xa) ;
dp dp =
2(x)dxadxa :
If the vector k (xa) is chosen to satisfy
then one nds
dX dX
= k2du2 + 2(p k)dudr + r2dp dp :
metric on I
+ with crosssectional metric
Since neither p nor k scales with r, the limit r ! 1 with u xed yields a degenerate
For instance, the at metric on Md corresponds to the choice
ds2 = r2 2(x) abdxadxb = r2ds2Md :
p (x) =
which yields the familiar coordinate transformation
and a metric of the form
X (u; r; xa) =
u + r(1 + x2)
; rxa; r(1
x2)
2
u
ds2 =
dudr + r2 abdxadxb :
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
HJEP05(218)6
In order to achieve cross sectional cuts of I+ which are metrically Sd, one chooses
p (x) =
2
The coordinate transformation is then and the spacetime metric is
X (u; r; xa) = u + r;
12+rxxa2 ; r
1
In order to make the relationship between Md and I
coordinate system (A.7). A massless eld in the plane wave basis takes the form
+ even more explicit, consider the at
"a11::::::ass (q) hOa1:::as (q)eiq X + Oay1:::as (q)e iq X i : (A.12)
so that X ! rp (xa). In this limit, the argument of the exponential
To perform an asymptotic analysis near I+, one considers the limit r ! 1 with u xed,
ir!qq(ya) p(xa) =
r!q(x
y)2
i
2
is large so that the exponential is rapidly oscillating. At leading order in 1r , the only
momenta that contribute to the integral are those for which the phase is stationary, i.e. for
which x = y. Therefore in the larger lightlike limit, one e ectively trades the transverse
coordinates on I
+ for the momentum coordinates on Md.
Armed with this knowledge we can further elaborate on the results of section 4. The
softphoton operator Sa(x) is related to the boundary current Ja(x) through a di erential
equation of the form
(
) d2 1Sa(x) = (4 )d=2 (d=2)Ja(x) :
(A.13)
(A.14)
The physical picture is clear: charged particles passing through I
+ act as a source Ja(x)
for the soft radiation Sa(x) (see [12] for relevant expressions relating (A.14) to the soft
charge for large U(1) gauge transformations). Similar statements apply to the gravitational
case [15]. Energetic particles passing through I
energy momentum tensor Tab) for softgraviton radiation Bab(x).
+ act as an e ective source (the boundary
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