Revisiting the Shadow Stress Tensor
in Celestial CFT

Shamik Banerjee1,2 and Sabrina Pasterski3

1 National Institute of Science Education and Research (NISER), Bhubaneswar
752050, Odisha, India

2 Homi Bhabha National Institute, Anushakti Nagar, Mumbai, India-400085

3 Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada

We revisit the standard construction of the celestial stress tensor as a
shadow of the subleading conformally soft graviton. In its original
formulation there is an obstruction to reproducing the expected $TT$ OPE in
the double soft limit. We propose a modification to the definition which
circumvents this obstruction and then extend this change of basis beyond the
conformally soft and single helicity sectors. In the process we investigate
how (non)-commutativity of double soft limits is tied to the decoupling of
primary descendants, and how our choice of celestial basis determines which
symmetries are manifest at the level of the OPE beyond the MHV sector.

###### Contents

  1. 1 Introduction
  2. 2 Symmetries of the Subleading Soft Graviton
  3. 3 A Modified ‘Shadow Basis’
  4. 4 Mixed Helicity Amplitudes: A Proposal
  5. 5 Discussion
  6. A Lessons from Gelfand
    1. A.1 Generalized Homogeneous Functions
    2. A.2 Representations at Integer Points
  7. B Complexifying the Celestial Sphere
  8. C OPE between modified stress tensors: $\bar{T}_{mod}\bar{T}_{mod}$

##  1 Introduction

Over the past few years celestial holographers have made progress towards
constructing a dual description for gravitational scattering in asymptotically
flat spacetimes [1]. Under this holographic map $\mathcal{S}$-matrix elements
are mapped to correlators in a conformal theory living on the celestial
sphere. The main advantage of this program is that it effectively reorganizes
amplitudes in terms of the available symmetries [2].

Lorentz invariance guarantees that a basis of boost eigenstates will transform
as quasi-primaries. In the simplest case of massless particles, we can reach
boost eigenstates from energy eigenstates by performing a Mellin transform in
the energies. The saddle point approximation tells us that when we push our
Cauchy slice to null infinity to prepare the in and out states, this procedure
picks out operators which are local on the celestial sphere but smeared along
the generators of $\mathcal{I}^{\pm}$. However, from the bulk perspective
there is some flexibility in our dictionary [3]. Namely, intertwiners like the
shadow transform map quasi-primaries to quasi-primaries.

A major motivation for pursuing the Celestial CFT (CCFT) construction comes
from the fact that the Lorentz symmetry is enhanced when we couple to gravity.
Intriguingly, a universal coupling to the subleading soft graviton [4] is
equivalent to the Ward identity for a Virasoro symmetry [5] and gives rise to
a candidate stress tensor [6]. The fact that we have such a stress tensor
hints that the dual might behave more like a local CFT than we would otherwise
have grounds to assert. Moreover, this would naïvely help fix the ambiguity in
our basis: the standard $T{\cal O}$ OPE is reproduced by the single subleading
soft graviton insertion, when the ${\cal O}$ operator is a ‘Mellin basis’
operator. However, $T$ itself is constructed as a shadow transform of the
$\Delta=0$ conformally soft graviton.

So which basis should we be using? When trying to phrase the soft physics in a
standard 2D CFT language, it appears that the natural dictionary for currents
[7, 8, 9, 10] involves shadow transforms of the conformally soft modes [11,
12, 13, 14, 15]. However, we see that the Mellin basis is the natural
dictionary for local operators capturing the finite energy radiative
scattering states, with a continuous spectrum on the principal series. It has
also been shown [16, 17] that in the standard Mellin basis the subleading soft
graviton theorem is equivalent to the Ward identity of
$\widehat{\overline{sl}}_{2}$ current algebra. This has been checked
explicitly by computing the celestial OPE of two positive helicity soft
gravitons in the MHV sector and this computation shows that the OPE can be
written as a linear combination of supertranslation and
$\widehat{\overline{sl}}_{2}$ current algebra descendants of positive helicity
gravitons. Meanwhile, the construction of $w_{1+\infty}$ generators from the
residues of the Mellin transformed amplitudes involves another intertwiner
that appears in split signature: the light transform [18, 19].

While the saddle point approximation is strictly only valid for the radiative
states, the standard procedure is to analytically continue off the principal
series [11, 20] to study behavior of celestial amplitudes in the
complex-$\Delta$ plane [21, 22]. In particular, we would like to be able to
continuously take the conformally soft limits of the $T{\cal O}$ OPE. In doing
so we encounter a variety of puzzles. Firstly, applying a shadow
transformation only in this limit appears ad hoc. Secondly, we run into
subtleties with double soft limits in either basis [23, 24, 8, 9, 25, 26],
which will be at the center of our explorations here. Thirdly, even after
resolving these issues, different basis choices make different symmetries
manifest at the level of the OPE.

More succinctly: in a 2D CFT an operator and its shadow cannot both be treated
as local operators. In the celestial context we have an additional constraint:
because the currents come from limits of the hard operators it seems like
either our dictionary should have all of the operators be shadowed or not.
This presents a tension between our dictionary of local operators in the 2D
theory, the existence of symmetry generators, and consistency of their
algebra.111It would be interesting to explore how this generalizes to higher
dimensions. Note that for $d>2$ we only need to demand that the operators are
quasi-primaries which will still be the case in the shadow basis [7, 10].
Focusing on the subleading conformally soft graviton sector, we can phrase the
puzzle as follows: We know that a positive helicity subleading soft graviton,
after shadow transformation, becomes the antiholomorphic stress tensor of a
$2$D (celestial) CFT. So we expect that we should be able to write the OPE in
the MHV sector as a linear combination of $\overline{Vir}$ primary and
descendants. But, this is not what we find. Rather, the OPE is written in
terms of supertranslation and $\widehat{\overline{sl_{2}}}$ current algebra
descendants. So a natural question is, what happens to the antiholomorphic
stress tensor? In this paper, we provide an answer to this question.

We begin by revisiting the standard construction of the celestial stress
tensor from [6], defined as a shadow of the subleading soft graviton[4, 5]. In
its original formulation, there is an obstruction to reproducing the expected
$TT$ OPE in the double soft limit considered in [8]. With the insights of the
celestial diamond and nested primary descendants [27, 28, 29, 30] this can be
cleanly phrased in terms of the weight $\Delta=-1$ reparameterization mode
from which both the celestial stress tensor and the subleading soft graviton
descend. Here, we propose a modification to the standard definition [6] which
circumvents this obstruction and then extend this change of basis beyond the
conformally soft and single helicity sector. This procedure is reminiscent of
monodromy projections familiar from standard CFT in the context of extracting
contributions from local operator exchanges to the conformal blocks [31, 32].
In the process, we show how the (non)-commutativity of double soft limits
signal obstructions to the decoupling of primary descendants in correlators,
as well as how our choice of basis determines which symmetries are manifest at
the level of the OPE.

This paper is organized as follows. In section 2 we revisit double soft limits
of the subleading soft graviton in the positive helicity sector, where we
observe an obstruction to the standard stress tensor Ward identity. We can
avoid this by introducing a modified shadow stress tensor. We then extend this
modification beyond the conformally soft sector in section 3 and to mixed
helicity amplitudes in section 4. Finally we conclude with a summary in
section 5. Some mathematical background is included in Appendix A, while we
make further contact with other incarnations of the celestial stress tensor in
Appendix B, before showing more explicitly how our modified stress tensor
avoids the aforementioned obstruction in Appendix C.

Before proceeding let us set up some notation. Unless otherwise specified, we
will consider massless scattering in (1,3) signature where the external
momenta take the form

$p_{i}=\epsilon_{i}\omega_{i}(1+z_{i}{\bar{z}}_{i},z_{i}+{\bar{z}}_{i},i({\bar{z}}_{i}-z_{i}),1-z_{i}{\bar{z}}_{i}),$
(1.1)

where $\epsilon_{i}=\pm 1$ and $\omega_{i}\geq 0$. Upon performing a Mellin
transform in the frequency variables $\omega_{i}$ the $\mathcal{S}$-matrix
gets mapped to a correlator of operators

$\prod_{i=1}^{n}\int
d\omega_{i}\omega_{i}^{\Delta_{i}-1}A(p_{i})=\langle\phi^{\epsilon_{1}}_{h_{1},{\bar{h}}_{1}}(z_{1},{\bar{z}}_{1})...\phi^{\epsilon_{n}}_{h_{n},{\bar{h}}_{n}}(z_{n},{\bar{z}}_{n})\rangle,$
(1.2)

which transform as quasi-primaries of weight
$h_{i}=\frac{1}{2}(\Delta_{i}+J_{i}),{\bar{h}}_{i}=\frac{1}{2}(\Delta_{i}-J_{i})$
under the Lorentz group. Here $J_{i}$ matches the helicity of the $i^{th}$
particle in an ‘all out’ convention. In Euclidean CFTs, the shadow transform
acts as an intertwiner between representations with Weyl reflected weights
$h_{i}\mapsto 1-h_{i},~{}~{}\bar{h}_{i}\mapsto 1-\bar{h}_{i}$. We will use
$\tilde{\phi}^{\epsilon_{i}}_{h_{i},{\bar{h}}_{i}}$ to denote the operators
reached by composing the Mellin transform (1.2) with the 2D shadow transform.
Unless necessary we will suppress the label $\epsilon_{i}$ on the external
operators.

##  2 Symmetries of the Subleading Soft Graviton

One feature of the celestial map (1.2) is that it sends powers of $\omega$ in
the soft expansion of gauge bosons to poles in the conformal dimension at
integer values of $\Delta$ when we analytically continue from the principal
series capturing radiative states to the complex plane. We will focus on soft
limits of the positive helicity graviton in this section. The subleading soft
graviton is picked out as the residue at $\Delta=0$ of the spin $h-\bar{h}=2$
operator $\phi_{h,\bar{h}}$ corresponding to the positive helicity gravitons

$S(z,{\bar{z}})=\mathrm{Res}_{\Delta=0}\phi_{h,\bar{h}}(z,{\bar{z}}).$ (2.1)

The soft theorem [4] tells us that this limit has a universal form in
celestial correlators

$\langle{S(w,\bar{w})\prod_{i=1}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle=-\sum_{k=1}^{n}\frac{\left(\bar{z}_{k}-\bar{w}\right)^{2}\bar{\partial}_{k}+2\bar{h}_{k}\left(\bar{z}_{k}-\bar{w}\right)}{w-z_{k}}\langle{\prod_{i=1}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle.$
(2.2)

The operator $S$ has weights $(1,-1)$. If we take its shadow we land on a
$(0,2)$ operator

$\bar{T}(z,\bar{z})=\frac{3}{\pi}\int
d^{2}w\frac{1}{\left(\bar{z}-\bar{w}\right)^{4}}S(w,\bar{w}),$ (2.3)

which was proposed as a candidate stress tensor in [6]. Indeed, so long as all
of the other operators are hard one can show that

$\langle{\bar{T}(z,\bar{z})\prod_{i=1}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle=\sum_{k}\left(\frac{\bar{h}_{k}}{\left(\bar{z}-\bar{z}_{k}\right)^{2}}+\frac{1}{\bar{z}-\bar{z}_{k}}\frac{\partial}{\partial\bar{z}_{k}}\right)\langle{\prod_{i=1}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle$
(2.4)

matching the expected conformal Ward identity. However, we must revisit our
construction if we want to be able to consistently take a second particle
soft.

As discussed in [33, 34] one expects the double soft limits of amplitudes in
the single helicity sector to commute, while this is not the case for
particles of opposite helicities.222This can happen for spin-0 excitations
when the conformal manifold has nontrivial curvature. See [35] for a nice
recent exploration of marginal deformations of celestial CFTs. The issue that
we run into here arises from the appearance of the shadow transformation in
our definition of $\bar{T}$. We can phrase this problem in an intrinsically 2D
language, and its resolution will be closely related to the monodromy
projections one needs to perform to select the stress tensor vs shadow blocks
[32] in ordinary, non-celestial, CFTs. Namely, equation (2.3) implies the
following relationship between primary333Here we are talking about primaries
of the global conformal group. descendants of $S$ and $\bar{T}$

$\partial\bar{T}(z,\bar{z})=-\frac{1}{2}\bar{\partial}^{3}S(z,\bar{z}).$ (2.5)

From the single soft theorem we see that these descendants vanish away from
other hard operator insertions, namely

$\langle{\partial\bar{T}(z,\bar{z})\cdots}\rangle=\text{Contact Terms}$ (2.6)

and

$\langle{\bar{\partial}^{3}S(z,\bar{z})\cdots}\rangle=\text{Contact Terms}.$
(2.7)

We want to emphasize that (2.5) holds at the level of the wavefunctions [36,
3, 28, 29] so whenever expressions (2.6) and (2.7) are valid, the contact
terms in (2.6) and (2.7) are related by the factor of $-2$ in (2.5). What will
break down, however, is whether the operator equations (2.6) and (2.7)
continue to hold in the presence of other soft insertions.

#### A symmetry-based argument

Before going into further detail, let us give a simple symmetry-based argument
for why (2.6) and (2.7) cannot hold simultaneously as operator equations. In
other words, why we expect $\bar{\partial}^{3}S(z,\bar{z})=0$ to be violated
in a 2D CFT with an antiholomorphic stress tensor. The argument goes as
follows: Assume that we can consistently make one of the positive helicity
hard gravitons in (2.4) subleading soft. This then implies that $S$ is a
$\overline{Vir}$ primary of weight $\bar{h}=-1$. While $\bar{\partial}^{3}S$
is a primary of the global conformal group but it is not a $\overline{Vir}$
primary. Therefore the equation $\bar{\partial}^{3}S=0$ is not
$\overline{Vir}$ invariant and cannot hold in our 2D CFT.444We can restore the
condition of being a Virasoro primary at the expense of introducing the
superrotation Goldstone mode via the Weyl covariant derivatives in [37, 38].
This vacuum structure becomes important at loop level [39, 40] but should not
appear in our discussion of the amplitudes at tree level, where the $\Delta=2$
mode is observed to decouple.

This observation has implications for the symmetries of scattering amplitudes
and the celestial CFT. For example, one can interpret the subleading soft
graviton theorem (2.2) as the Ward identity for $\widehat{\overline{sl}}_{2}$
current algebra [16] and $S(z,\bar{z})$ as the generating function for the
corresponding $\widehat{\overline{sl}}_{2}$ currents. Our observation,
together with (2.5), leads to the conclusion that

If we admit both $S(z,\bar{z})$ and its shadow $\bar{T}(z,\bar{z})$ as local
operators in the theory then both the $\widehat{\overline{sl}}_{2}$ current
algebra and the $\overline{Vir}$ symmetries will be violated.

As a result, the whole $w_{1+\infty}$ tower of symmetries [19] will be also be
absent from the celestial theory. We will discuss this and other consequences
later in the paper.

#### Shadows and Double Soft Limits

Let us now explicitly show that (2.5) implies an inconsistency of combining
consecutive soft limits with the shadow operation and expecting a conservation
law for tree level amplitudes. Starting from the stress tensor insertion
(2.4), if we now send the conformal dimension of one of the hard particles to
$0$ we see that the form of the prefactor on the right hand side of (2.4)
implies there will be non-contact terms in the following level-3 descendant

$\displaystyle\langle{\bar{T}(z,\bar{z})\bar{\partial}^{3}S(w,{\bar{w}})\prod_{i=2}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle$
(2.8)
$\displaystyle=\left(-\frac{24}{\left(\bar{z}-\bar{w}\right)^{5}}-\frac{12}{\left(\bar{z}-\bar{w}\right)^{4}}\frac{\partial}{\partial\bar{w}}\right)\langle
S(w,{\bar{w}}){\prod_{i=2}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle+\text{Contact
Terms},$

where the prefactor comes from computing

$\left[\partial_{\bar{w}}^{3},\left(\frac{-1}{\left(\bar{z}-\bar{w}\right)^{2}}+\frac{1}{\bar{z}-\bar{w}}\frac{\partial}{\partial\bar{w}}\right)\right]=\left(-\frac{24}{\left(\bar{z}-\bar{w}\right)^{5}}-\frac{12}{\left(\bar{z}-\bar{w}\right)^{4}}\frac{\partial}{\partial\bar{w}}\right)$
(2.9)

and we have used $\bar{h}=-1$ for the operator $S$. We know from (2.2) that
this correlator has support for $(w,{\bar{w}})$ away from other operator
insertions. Namely the primary descendant $\bar{\partial}^{3}S$ no longer
vanishes as an operator in the presence of a celestial stress tensor
insertion. Using our descendancy identity (2.5) we see that

$\displaystyle\langle{\bar{T}(z,\bar{z})\partial\bar{T}(w,{\bar{w}})\prod_{i=2}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle$
(2.10)
$\displaystyle=\left(\frac{12}{\left(\bar{z}-\bar{w}\right)^{5}}+\frac{6}{\left(\bar{z}-\bar{w}\right)^{4}}\frac{\partial}{\partial\bar{w}}\right)\langle
S(w,{\bar{w}}){\prod_{i=2}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle+\text{Contact
Terms}.$

The presence of this non-contact term serves as an obstruction to the OPE for
$\bar{T}$ that would be expected from the BPZ construction. By contrast,
repeating the same manipulations starting from (2.4) we would run into no such
obstruction to

$\displaystyle\langle{S(z,\bar{z})\bar{\partial}^{3}S(w,{\bar{w}})\prod_{i=2}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle=\text{Contact
Terms}$ (2.11)

and similarly

$\displaystyle\langle{S(z,\bar{z})\partial\bar{T}(w,{\bar{w}})\prod_{i=2}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle=\text{Contact
Terms}$ (2.12)

consistent with what we expect from the fact that soft limits of same helicity
gravitons commute and the descendancy relation (2.5), though this will no
longer be the case if we also include the opposite helicity soft sector.
Equation (2.11) tells us we can construct a consistent
$\widehat{\overline{sl}}_{2}$ current algebra from the positive helicity
subleading soft graviton while equation (LABEL:TdecT) tells us that the shadow
stress tensor $\bar{T}$ does not generate a consistent $\overline{Vir}$
symmetry as written, due to an obstruction that arises when more than one
operator goes soft.

#### A Modified Stress Tensor

In the remainder of this section we will show that we can provide a
modification to our definition for the stress tensor so that we get a
consistent $\overline{Vir}$ symmetry. To do so we will take inspiration from
the ‘celestial diamond’ framework of [30, 29] and discussion of gauge
redundancies amongst nested primary descendants in [27, 28]. Namely, we will
start by lifting both $S$ and $\bar{T}$ to a weight $(0,-1)$ operator
$\epsilon$

$S=\partial\epsilon,~{}~{}~{}\bar{T}=-\frac{1}{2}\bar{\partial}^{3}\epsilon$
(2.13)

whose correlators take the form

$\displaystyle\langle{\epsilon(z,\bar{z})\prod_{i=1}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle$
(2.14)
$\displaystyle=-\sum_{k=1}^{n}\left(\ln\mu^{2}|z-z_{k}|^{2}\right)\\{\left(\bar{z}_{k}-\bar{z}\right)^{2}\bar{\partial}_{k}+2\bar{h}_{k}\left(\bar{z}_{k}-\bar{z}\right)\\}\langle{\prod_{i=1}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle,$

where $\mu$ is an IR regulator of the 2D model. Note that the $\mu$-dependence
drops out because of the global Ward identity

$\sum_{k=1}^{n}\\{\left(\bar{z}_{k}-\bar{z}\right)^{2}\bar{\partial}_{k}+2\bar{h}_{k}\left(\bar{z}_{k}-\bar{z}\right)\\}\langle{\prod_{i=1}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle=0.$
(2.15)

One can see that (2.14) reproduces both (2.2) and (2.4) upon taking the
appropriate derivatives (2.13). Moreover (2.5) follows automatically from
(2.13). The operator $\epsilon$ has the same weights as a reparameterization
mode. While we will postpone this interpretation for the time being, the
manipulations surrounding monodromy projections and distinguishing between
local operator versus shadow exchanges, as encountered in [32], are relevant
to this story.555In the celestial context there are naturally two
symplectically paired ‘memory’ and ‘Goldstone’ modes. The diamond descending
from $\epsilon$ measures the spin memory effect [41], while the paired
superrotation Goldstone mode is the more natural ‘reparameterization’ operator
[42, 43] and descends to its own diamond of SL$(2,\mathbb{C})$ primaries.
Focusing on the memory modes, if we consider splitting the correlator (2.14)
into two parts by replacing
$\log\mu^{2}|z-z_{k}|^{2}=\log\mu(z-z_{k})+\log\mu({\bar{z}}-{\bar{z}}_{k}),$
(2.16) we see that the presence of both terms guarantees the $S$ and its
shadow $\bar{T}$ both descend from this top corner of the diamond. With either
the first or second term on its own, we can essentially kill either the left
or right corner of the celestial diamond. This different projections can be
phrased as gauging one of two transformations [32]
$[\epsilon(z,{\bar{z}})]_{S}=\epsilon(z,{\bar{z}})+\Lambda({\bar{z}}),~{}~{}~{}~{}[\epsilon(z,{\bar{z}})]_{\bar{T}}=\epsilon(z,{\bar{z}})+\Lambda_{0}(z)+\Lambda_{1}(z){\bar{z}}+\Lambda_{2}(z){\bar{z}}^{2}.$
(2.17) However, the additional subtlety we encounter here is that doing such a
monodromy projection of the correlator not only impacts our prescription for
both the $\epsilon$ operator explicitly shown in (2.14) but also the hard
operators whose conformally soft limits are related to $\epsilon$ by (2.13).

In the celestial context we are also interested in understanding the correct
external operators rather than just how to project their correlators onto
‘physical’ exchanges when we decompose them into lower point amplitudes. This
lends itself to a slightly different phrasing of the problem. As we will
discuss in the next section, being able to consistently take the conformally
soft limit implies a similar restriction on what hard operators we can insert.
We can choose a different Green’s function so that

$S=\partial\epsilon_{S},~{}~{}~{}\langle\bar{\partial}^{3}\epsilon_{S}(w,{\bar{w}})\prod_{i=1}^{n}\phi_{h_{i},{\bar{h}}_{i}}(z_{i},{\bar{z}}_{i})\rangle=\text{Contact
Terms},$ (2.18)

where the left hand equation holds away from other operator insertions and the
right hand equation assumes all other operator insertions are hard. More
concretely, assuming that we can analytically continue our expression for the
soft theorem (2.2) off of the celestial sphere $z^{*}={\bar{z}}$, we can
define following non-local operator

$\epsilon_{S}=\int_{z_{0}}^{z}dwS(w,{\bar{z}}),$ (2.19)

where the open contour runs between a reference point, $z_{0}$, and $z$. Since
$S$ has weight $h=1$ the integrand has the appropriate left-handed weight. The
correlation function of $\epsilon_{S}$ is given by

$\displaystyle\langle{\epsilon_{S}(z,\bar{z})\prod_{i=1}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle$
(2.20)
$\displaystyle=-\sum_{k=1}^{n}\ln\mu\left(z-z_{k}\right)\\{\left(\bar{z}_{k}-\bar{z}\right)^{2}\bar{\partial}_{k}+2\bar{h}_{k}\left(\bar{z}_{k}-\bar{z}\right)\\}\langle{\prod_{i=1}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle.$

Now defining $\bar{T}$ via the usual shadow definition (2.3), we can construct
the following candidate for a modified stress tensor666Note that it is
compatible to view our correction term as removing a contribution from the
hard charge, a weight $\Delta=3$, $J=-1$ operator constructed from the matter
fields
$\partial\bar{T}=j_{3,-1}~{}~{}\Leftrightarrow~{}~{}\partial(\bar{T}-\partial^{-1}j_{3,-1})=0,$
(2.21) so that the Ward identity enforces conservation of the modified stress
tensor, identically [44].

$\bar{T}_{\rm
mod}=\bar{T}+\frac{1}{2}\bar{\partial}^{3}\epsilon_{S},~{}~{}~{}\partial\bar{T}_{\rm
mod}=0.$ (2.22)

It’s conservation follows from (2.5) and the first equation in (2.18), while
the fact that it reduces to $\bar{T}$ in the absence of other soft insertions
follows from the second equation in (2.18). If we repeat the same
manipulations that led to the obstruction (LABEL:TdecT) with $\bar{T}$
replaced with $\bar{T}_{\rm mod}$, we indeed find that the non-contact terms
vanish, as expected from the operator equation (2.18).777See appendix C for
the details. However the analog of (LABEL:Sdec) implies that
$\bar{\partial}^{3}S$ is non-vanishing in the presence of a $\bar{T}_{\rm
mod}$ insertion so, much like the monodromy projection description, you give
up the $\widehat{\overline{sl}}_{2}$ symmetry to get a manifest
$\overline{Vir}$ symmetry.

##  3 A Modified ‘Shadow Basis’

The essence of our construction of $T_{mod}$ boiled down to exploiting an
ambiguity in the Green’s functions lifting us from the subleading soft
graviton to the reparameterization mode. This nested structure of primary
descendants is known to persist for other conformally soft operators with
(negative) integer weights. While we now have a prescription for the stress
tensor that reproduces the correct (centerless) Virasoro Ward identity for
both single and double soft insertions, there is still a tension with the fact
that all of the other operators are un-shadowed. Even at the level of matching
the single soft insertions, we are not shadow transforming all operators
simultaneously, but constructing a highly non-local object only for
conformally soft modes. In this section we discuss how to generalize our
construction of $T_{mod}$ beyond the subleading conformally soft limit to
general complex weights.

Let’s start with the integer point case first. We will focus on the
holomorphic sector here, though an analogous discussion holds for the anti-
holomorphic case. See also the discussion in Appendix (A). Suppose
$\phi_{h}(z)$ is a SL$(2,\mathbb{C})$ primary. Namely,

$L_{0}\phi_{h}=h\phi_{h},\ L_{1}\phi_{h}=0.$ (3.1)

The representation is spanned by the descendants $L_{-1}^{n}\phi_{h}$. A
‘primary descendant’ will be a state in this module such that

$\psi_{n}=L_{-1}^{n}\phi_{h},~{}~{}~{}L_{1}\psi_{n}=0.$ (3.2)

Applying the standard commutation relations we get

$L_{1}\psi_{n}=n\left(n+2h-1\right)L_{-1}^{n-1}\phi_{h}.$ (3.3)

Therefore a non-trivial ‘primary descendant’ exists for the unique value

$n=1-2h.$ (3.4)

Now, in terms of the fields, we can write

$\psi_{n}(z)=\partial^{1-2h}\phi_{h}(z).$ (3.5)

Since $h$ takes on the special (half) integer value (3.4), this is an ordinary
derivative. However, we can try to analytically continue this to an arbitrary
complex number $h$.

#### An Integral Representation

Let us consider the differential equation

$\frac{d^{n}}{dz^{n}}g_{n}(z)=g(z).$ (3.6)

One solution to (3.6) can be written as

$g_{n}(z)=\frac{1}{\Gamma(n)}\int_{a}^{z}dw(z-w)^{n-1}g(w).$ (3.7)

Here $g_{n}(z)$ represents the $n$-fold integral of the function $g(z)$.
Formally we can invert this by taking $n\mapsto-n$ in which case the $n$-fold
derivative of the function $g(z)$ is given by

$\frac{d^{n}}{dz^{n}}g(z)=\lim\limits_{\epsilon\rightarrow
0}\frac{1}{\Gamma(\epsilon-n)}\int_{a}^{z}dw(z-w)^{\epsilon-n-1}g(w).$ (3.8)

Here $n$ is still an integer. Analytically continuing this to general $n=1-2h$
we can formally write

$\partial^{1-2h}\phi_{h}(z)=\frac{1}{\Gamma(2h-1)}\int_{z_{0}}^{z}\frac{dw}{(z-w)^{2-2h}}\phi_{h}(w)\equiv\phi^{\prime}_{1-h}(z_{0},z).$
(3.9)

The choice of a reference point $z_{0}$ explicitly breaks the expected global
conformal covariance, since this is clearly no longer a local operator. It
will be convenient to take our reference point $z_{0}\rightarrow\infty$. While
the point at infinity is not preserved by the global conformal group, we do
have some control over the behavior of the scattering amplitudes at large
angular separation. Indeed take $z_{0}\rightarrow\infty$ keeping $z$ fixed we
expect $\phi_{h}(z_{0})\rightarrow\frac{1}{z_{0}^{2h}}$. Meanwhile,

$\frac{\partial}{\partial
z_{0}}\phi^{\prime}_{1-h}(z_{0},z)=-\frac{1}{\Gamma(2h-1)}\frac{\phi_{h}(z_{0})}{(z-z_{0})^{2-2h}}$
(3.10)

so the leading term in $z_{0}$ is given by

$\frac{\partial}{\partial
z_{0}}\phi^{\prime}_{1-h}(z_{0},z)\sim\frac{1}{z_{0}^{2}}\rightarrow 0,\
z_{0}\rightarrow\infty.$ (3.11)

It is natural to call (3.9) an ‘incomplete light ray operator.’ We see that
the integrand is the same (and is the holomorphic ‘half’ of the shadow
kernel). As compared to the ones appearing in [45, 46, 47] it extends over
half the range.

#### Celestial Diamond Redux

Now let us see how these appear in our construction of a modified shadow
basis. We will need to restore the dependence on ${\bar{z}}$. Consider a
primary field $\phi_{h,\bar{h}}$ and its shadow
$\tilde{\phi}_{1-h,1-\bar{h}}$. For generic weights we will define our shadow
transform as follows

$\displaystyle\widetilde{\mathcal{O}_{h,\bar{h}}}(w,\bar{w})=\frac{K_{h,\bar{h}}}{2\pi}\int
d^{2}w^{\prime}\frac{\mathcal{O}_{h,\bar{h}}(w^{\prime},{\bar{w}}^{\prime})}{(w-w^{\prime})^{2-2h}({\bar{w}}-{\bar{w}}^{\prime})^{2-2\bar{h}}}\,.$
(3.12)

The normalization constant $K_{h,\bar{h}}$ was chosen to be
$K_{h,\bar{h}}=h+\bar{h}+|h-\bar{h}|-1$ in [3, 29] so that
$\widetilde{\widetilde{\mathcal{O}_{h,\bar{h}}}}=(-1)^{2(h-\bar{h})}\mathcal{O}_{h,\bar{h}}$,
matching the conventions of [48]. Meanwhile
$K_{h,\bar{h}}=\frac{\Gamma(2-2\bar{h})}{\Gamma(2h-1)}$ in [49]. We can leave
it unspecified for the time being, so as to keep track of which statements are
independent of this convention.

To connect to our discussion above, we need to understand the limiting
behavior near integer points. The identity
$\partial_{z}\frac{1}{\bar{z}}=\pi\delta^{(2)}(z)$ (matching the conventions
of [49]) tells us that for special integer weights the shadow kernel becomes a
Green’s function [29], namely

$\partial_{{\bar{w}}^{\prime}}^{\bar{k}}\frac{({\bar{w}}^{\prime}-{\bar{w}})^{\bar{k}-1}}{(w^{\prime}-w)^{k+1}}=\pi(\bar{k}-1)!\frac{(-1)^{k}}{k!}\partial_{w^{\prime}}^{k}\delta^{(2)}(w^{\prime}-w)\,,$
(3.13)

and similarly for the holomorphic sector for $k$ and $\bar{k}$ flipped.
Meanwhile from Gelfand [50] we know that

$\left[\frac{z^{\lambda}{\bar{z}}^{\mu}}{\Gamma\left(\frac{1}{2}(\mu+\lambda)+\frac{1}{2}|\mu-\lambda|+1\right)}\right]_{\overset{\lambda=-k-1}{\mu=-l-1}}=\pi\frac{(-1)^{k+l+j}j!}{k!l!}\delta^{(k,l)}(z,{\bar{z}}),$
(3.14)

where $j=\frac{1}{2}(\mu+\lambda)-\frac{1}{2}|\mu-\lambda|=\min\\{k,l\\}$. We
will explore these integer point representations in more detail in Appendix A;
however, the essence of the celestial diamond story we need here is that the
shadow kernel can be interpreted as either Green’s function or differential
operators taking us between conformal primaries with Weyl reflected weights
[29, 30] (see also [20]). We can carry these observations away from the
integer points by treating the shadow and light transforms as pseudo-
differential operators.

The pseudo-differential operators introduced above are such that as we limit
to integer points we have the following descendancy relation

$(-1)^{2h-1}\Gamma(2-2h)\partial^{2h-1}\tilde{\phi}_{1-h,1-\bar{h}}=K_{h,\bar{h}}\Gamma(2\bar{h}-1)\bar{\partial}^{1-2\bar{h}}\phi_{h,\bar{h}}$
(3.15)

generalizing (2.5). The analog of (2.18) is to introduce a field
$f_{1-h,\bar{h}}$ such that

$\displaystyle\phi_{h,\bar{h}}=(-1)^{2h-1}\Gamma(2-2h)\partial^{2h-1}f_{1-h,\bar{h}},~{}~{}~{}\langle\bar{\partial}^{1-2\bar{h}}f_{1-h,\bar{h}}\prod_{i=1}^{n}\phi_{h_{i},{\bar{h}}_{i}}(z_{i},{\bar{z}}_{i})\rangle=\text{Contact
Terms}$

(3.16)

so that the analog of (2.22) is

$\tilde{\phi}_{1-h,1-\bar{h};mod}=\tilde{\phi}_{1-h,1-\bar{h}}-K_{h,\bar{h}}\Gamma(2\bar{h}-1)\bar{\partial}^{1-2\bar{h}}f_{1-h,\bar{h}},$
(3.17)

while (3.15) guarantees the analog of the conservation law
$\partial\bar{T}_{mod}=0$, implying the primary descendants that appear at
integer points decouple. In terms of our explicit integral expressions for
these pseudo differential operators we have

$\scalebox{0.98}{\mbox{$\displaystyle\tilde{\phi}_{1-h,1-\bar{h};mod}=\frac{K_{h,\bar{h}}}{2\pi}\left[\int_{\hat{\mathbb{C}}}d^{2}w+\frac{i}{2}(1-e^{2\pi
i(2-2h)})\int_{-\infty}^{z}dw\int_{-\infty}^{\bar{z}}d{\bar{w}}\right]\frac{{\phi}_{h,\bar{h}}(w,{\bar{w}})}{(z-w)^{2-2h}({\bar{z}}-{\bar{w}})^{2-2{\bar{h}}}}$}},$
(3.18)

where

$K_{h,\bar{h}}=\frac{\Gamma(2-2\bar{h})}{\Gamma(2h-1)}=\frac{\Gamma(2-2h)}{\Gamma(2\bar{h}-1)},~{}~{}~{}h-\bar{h}=\pm
2.$ (3.19)

The formula is the same for both positive and negative helicity
gravitons.888For $\Delta=1-|J|-n$ for $n\in\mathbb{Z}_{>}$ we expect both
holomorphic and antiholomorphic primary descendants from the representation
theory, however we see that this would appear to involve an additional
subtraction as compared to (3.18).

We see that if we complexify the shadow integral kernel, for generic weights
there is a branch cut that extends from $w=z$ to $w=\infty$ in the complex $w$
plane and similarly, from ${\bar{w}}={\bar{z}}$ to ${\bar{w}}=\infty$ in the
complex ${\bar{w}}$ plane. The monodromy around $z=w$ is the phase $e^{2\pi
i(2h-2)}$. Taking this into account, we can recognize the normalization as
arising from the discontinuity across the branch cut in the complex $w$ plane
for a particular orientation of the contour. We won’t pursue this
interpretation further here, but will revisit the complexification of the
celestial sphere to $\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ in appendix B in
the context of comparing our subtraction to other constructions in the
literature.

Now we’ve seen that by construction $\partial^{2h-1}\tilde{\phi}_{mod}$
vanishes in correlators. Because of (3.18) and (3.15) this is true both for
contact terms as well so we are free to smear our correlators. Rather than
restrict to the other operators being of $\phi_{h,{\bar{h}}}$ type, they can
also be shadowed or modified shadow operators

$\langle\partial^{2h-1}\tilde{\phi}_{1-h,1-\bar{h};mod}\tilde{\phi}_{1-h_{i},1-\bar{h}_{i};mod}\ldots\rangle={\rm
Contact~{}Terms}.$ (3.20)

In particular, for $h=1$ and $\bar{h}=-1$ this reduces to the conservation law
for $\bar{T}_{mod}$ in these modified correlators

$\langle\partial\bar{T}_{mod}\tilde{\phi}_{1-h_{i},1-\bar{h}_{i};mod}\ldots\rangle={\rm
Contact~{}Terms}.$ (3.21)

In these expressions we’ve assumed that the $...$ included no other soft
limits of the opposite helicity sector. We will now turn to the generic mixed
helicity case.

##  4 Mixed Helicity Amplitudes: A Proposal

Let us now go beyond the single helicity sector. The conventional double soft
theorems for positive and negative helicity gravitons are known to have
ambiguities [34]. Here we will see that sequential conformally soft limits
present the same type of obstruction to the decoupling of primary descendants.
Again we will start by focusing on the subleading soft graviton. Let $\bar{S}$
denote the $\Delta=0,J=-2$ soft graviton extracted as in (2.1) above, but for
the negative helicity sector. If we consider a mixed helicity correlator and
take a sequential conformal soft limit where the $+$ helicity operator is
taken to $\Delta=0$ first, we find

$\begin{gathered}\langle{S(w_{1},\bar{w}_{1})\bar{S}(w_{2},\bar{w}_{2})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle_{-+}\\\
=-\frac{\left(\bar{w}_{2}-\bar{w}_{1}\right)^{2}\bar{\partial}_{w_{2}}+2\left(\bar{w}_{2}-\bar{w}_{1}\right)}{w_{1}-w_{2}}\langle{\bar{S}(w_{2},\bar{w}_{2})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle\\\
-\sum_{i}\frac{\left(\bar{z}_{i}-\bar{w}_{1}\right)^{2}\bar{\partial}_{i}+2\bar{h}_{i}\left(\bar{z}_{i}-\bar{w}_{1}\right)}{w_{1}-z_{i}}\langle{\bar{S}(w_{2},\bar{w}_{2})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle,\end{gathered}$
(4.1)

while taking the opposite order gives

$\begin{gathered}\langle{S(w_{1},\bar{w}_{1})\bar{S}(w_{2},\bar{w}_{2})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle_{+-}\\\
=-\frac{\left(w_{1}-w_{2}\right)^{2}\partial_{w_{1}}+2\left(w_{1}-w_{2}\right)}{\bar{w}_{2}-\bar{w}_{1}}\langle{S(w_{1},\bar{w}_{1})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle\\\
-\sum_{i}\frac{\left(z_{i}-w_{2}\right)^{2}\partial_{i}+2h_{i}\left(z_{i}-\bar{w}_{2}\right)}{\bar{w}_{2}-\bar{z}_{i}}\langle{S(w_{1},\bar{w}_{1})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle.\end{gathered}$
(4.2)

While one can further plug in the form of the soft theorem to evaluate the
remaining correlators in these expressions and see that
$(\ref{plus})\neq(\ref{minus})$, we’ve paused here because we already
encounter an obstruction to the null state decoupling we’d expect. Namely,
from the single soft limits we expect the primary descendants
$\bar{\partial}^{3}S$ and $\partial^{3}\bar{S}$ to decouple. However we see
that in (4.1) we have $\bar{\partial}^{3}S(z,\bar{z})=0$ but
$\partial^{3}\bar{S}(z,\bar{z})\neq 0$, while the opposite is true for (4.2).

Any attempt to try to evade the ambiguity by specifying a priori which order
of soft limits to take still comes at a cost. In the $(-+)$ case we preserve
the $\widehat{\overline{sl}}_{2}$ current algebra but lose the
$\widehat{sl}_{2}$ current algebra. Similarly in the $(+-)$ case the
$\widehat{sl}_{2}$ current algebra still holds but we lose the
$\widehat{\overline{sl}}_{2}$ symmetry. The fact that we are unable to
preserve both decoupling conditions at the same time is perhaps not as
surprising given that, taken together, the two current algebras do not form a
closed algebra. Meanwhile we don’t have obvious candidates for the additional
generators of Diff$(S^{2})$ (see also [51]). Phrased in this manner, we
recognize this as the same type of problem that we started with in the single
helicity sector. Namely, just as we saw from the double soft theorems that we
couldn’t consistently realize the $\overline{Vir}$ and the
$\widehat{\overline{sl}}_{2}$ current algebra symmetries at the same, we
similarly need to look for additional generators since the
$\widehat{\overline{sl}}_{2}$ current algebra and $\overline{Vir}$ do not form
a closed algebra.

By contrast the $\overline{Vir}$ and $\widehat{sl}_{2}$ generators form a
closed algebra. The same is true for $Vir$ and $\widehat{\overline{sl}}_{2}$
(and also for $Vir\times\overline{Vir}$). Can we use this to augment our soft
algebras beyond the single helicity sector? Let’s return for a moment to the
expression for the $(-+)$ limit given by (4.1), and now do a shadow
transformation of $S(w_{1},\bar{w}_{1})$

$\begin{gathered}\langle{\bar{T}(w_{1},\bar{w}_{1})\bar{S}(w_{2},\bar{w}_{2})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle_{-+}\\\
=\left(\frac{1}{\left(\bar{w}_{1}-\bar{w}_{2}\right)^{2}}+\frac{1}{\bar{w}_{1}-\bar{w}_{2}}\frac{\partial}{\partial\bar{w}_{2}}\right)\langle{\bar{S}(w_{2},\bar{w}_{2})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle\\\
+\sum_{i}\left(\frac{\bar{h}_{i}}{\left(\bar{w}_{1}-\bar{z}_{i}\right)^{2}}+\frac{1}{\bar{w}_{1}-\bar{z}_{i}}\frac{\partial}{\partial\bar{z}_{i}}\right)\langle{\bar{S}(w_{2},\bar{w}_{2})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle.\end{gathered}$
(4.3)

We see that the problem we encountered above goes away, namely
$\partial^{3}\bar{S}=0$ and $\partial\bar{T}=0$. One can check that the same
is true if we started with the opposite choice $(+-)$ of conformally soft
limits. Now of course from the previous section we know that we want to
promote $\bar{T}\mapsto\bar{T}_{mod}$ to get a consistent $\overline{Vir}$
algebra but those statements were within that single helicity sector.

Taking a step back, we have seen that consecutive soft limits in the mixed
helicity sector run into a problem because of the fact that we have both
holomorphic and antiholomorphic poles. Equation (4.3) suggests that one way to
resolve this issue will be to consider, instead of the standard [52] celestial
amplitudes, amplitudes of the form

$\langle{+++\ldots\widetilde{-}_{mod}\widetilde{-}_{mod}\widetilde{-}_{mod}\ldots}\rangle$
(4.4)

and

$\langle{---\ldots\widetilde{+}_{mod}\widetilde{+}_{mod}\widetilde{+}_{mod}\ldots}\rangle$
(4.5)

where one helicity sector has been mod-shadowed before taking any soft limit.

In the first case, we expect that the symmetry that is realized locally is at
least as big as the semi-direct product of ${Vir}$ and $w_{1+\infty}$ and,
similarly, in the second case it is the semi-direct product of
${\overline{Vir}}$ and $\overline{w}_{1+\infty}$. In this paper, while we will
not try to give a proof of this proposal, we end this section by pointing out
some of its novel features.

The main point of our proposal is that the symmetry that is locally realized
at the level of $\cal S$-matrix depends on the choice of basis for the
asymptotic states. This may not be very surprising given that the basis
transformation which takes us from (4.4) to (4.5) is non-local. In one
description we declare the mod-shadowed operators $\widetilde{+}_{mod}$ as
local and in the other description, the conventional $+$ operators are local.
But, there is no useful description in which both the $+$ and
$\widetilde{+}_{mod}$ coexist as local operators. This is consistent with our
observation in section-$2$ that if we admit both $S(z,\bar{z})$ and
$\bar{T}(z,\bar{z})=\tilde{S}(z,\bar{z})$ as local operators in the theory
then we lose both the $\widehat{\overline{sl_{2}}}$ and $\overline{Vir}$
symmetries.

A relatively simple check of our proposal can be made in the MHV sector. First
of all, it was shown in [53] that the MHV sector has a symmetry group which is
the semidirect product of $Vir$ and $w_{1+\infty}$. This is consistent with
our proposal here. Furthermore, we know that when the MHV sector is described
by the standard amplitudes $\langle{--+++\cdots+}\rangle$, the celestial OPE
of two gravitons can be written [16] in terms of supertranslation and
$\widehat{\overline{sl}}_{2}$ current algebra descendants. According to our
proposal if instead, we describe the MHV sector by amplitudes of the form
$\langle{--\widetilde{+}_{mod}\widetilde{+}_{mod}\cdots\widetilde{+}_{mod}}\rangle$
then one should be able to write the celestial OPE between two gravitons as a
linear combination of supertranslation and $\overline{Vir}$ descendants. This
will be a nontrivial check of our proposal and we hope to return to some of
these problems in near future.

##  5 Discussion

Since the early days of celestial CFT, there has been an underlying debate
about which scattering basis in the bulk should correspond to the local
operators in the celestial dictionary. From the point of view of the global
Lorentz symmetry any intertwiner between primaries should give an equivalent
basis. However, the extrapolate dictionary [30] seems to prefer the Mellin
transformed basis because of the way that the momentum space and position
space celestial spheres get identified under the large-$r$ saddle point near
null infinity [54]. The enhancement from global SL$(2,\mathbb{C})$ to a
Virasoro symmetry would seem to settle this debate if it were not for the fact
that the symmetry generators are constructed from a non-local shadow
transform, while their Ward identities transforming hard operators demand that
those are defined in terms of the ordinary Mellin transformed basis.

Here we’ve shown that there is a tension with the definition of the stress
tensor as the shadow operator and shown how to remedy it. The underlying
reason for this tension is more general than our CCFT application: an operator
and its shadow can’t both be local. However the fact that we are necessarily
confronted with it is unique to the celestial case. We cannot separately
prescribe different dictionaries for the hard and soft modes without being
able to interpolate between the two when we continuously deform our hard
operator dimensions to various conformally soft limits. In particular, the
Virasoro symmetry picks out local operator for hard particles but wants to
have its cake and eat it to with the soft graviton mode defining the stress
tensor.

Our modification involves a certain projection of one half of the celestial
diamonds so that there is only one radiative soft mode. We then showed how to
extend this to operators of any weight, so that we can go to a basis where the
corrected shadow transform is again a conformally soft limit. In the end,
we’ve seen that different bases make different symmetries manifest at the
level of the celestial OPE and explored how certain helicity-asymmetric
representations of the amplitudes can be better suited to taking multi-soft
limits. These explorations expand upon a variety of exciting approaches to
elucidate the symmetries and structure of the 4D $\mathcal{S}$-matrix [55, 16,
8, 18, 19, 46, 38, 20, 51, 56].

### Acknowledgements

We would like to thank Yangrui Hu, Ashoke Sen, Atul Sharma, Andrew Strominger,
Tomasz Taylor and Herman Verlinde for useful conversations. The work of SB is
partially supported by the Swarnajayanti Fellowship (File No-
SB/SJF/2021-22/14) of the Department of Science and Technology and SERB, India
and by SERB grant MTR/2019/000937 (Soft-Theorems, S-matrix and Flat-Space
Holography). SP is supported by the Celestial Holography Initiative at the
Perimeter Institute for Theoretical Physics and has been supported by the Sam
B. Treiman Fellowship at the Princeton Center for Theoretical Science.
Research at the Perimeter Institute is supported by the Government of Canada
through the Department of Innovation, Science and Industry Canada and by the
Province of Ontario through the Ministry of Colleges and Universities.

##  Appendix A Lessons from Gelfand

In this appendix we briefly summarize some results from volumes 1 and 5 of
Gelfand’s series on generalized functions [50, 57]. In particular, we will
want to review his construction of homogeneous generalized functions and the
behavior of our SL$(2,\mathbb{C})$ primaries near integer conformal dimensions
so that we can understand various contact terms that appear and how to handle
their analytic continuations to other signatures.

###  A.1 Generalized Homogeneous Functions

Appendix B of Gelfand Volume 1 [50] discusses generalized functions of a
complex variable. These are relevant for our understanding of the
distributions that appear when we take descendants at special values of the
conformal dimension. In particular our mode expansion of the conformal
primaries involves so-called homogeneous generalized functions, whose
definitions and properties we review here.

Following [50], a function $F(z,{\bar{z}})$ is a homogeneous function of
degree $(\lambda,\mu)$ if for $a\in\mathbb{C}-\\{0\\}$

$F(az,\bar{a}{\bar{z}})=a^{\lambda}\bar{a}^{\mu}F(z,{\bar{z}}).$ (A.1)

This can be promoted to a generalized function by considering the following
functional

$(F,\varphi)=\int d^{2}zF(z,{\bar{z}})\varphi(z,{\bar{z}})$ (A.2)

for an appropriate set of test functions $\varphi$ with bounded support so
that we are free to integrate by parts

$(f,\varphi^{(j,k)})=(-1)^{j+k}(f^{(j,k)},\varphi).$ (A.3)

In particular, via a change of variables we see that for our homogeneous
functions we have

$(F,\varphi(z/a,{\bar{z}}/\bar{a}))=a^{\lambda+1}\bar{a}^{\mu+1}(F,\varphi).$
(A.4)

For ${\rm Re}(\mu+\lambda)>-2$ we can defined the generalized function
$z^{\lambda}{\bar{z}}^{\mu}$ via the functional
$(z^{\lambda}{\bar{z}}^{\mu},\varphi)$ which converges for $\varphi$
infinitely differentiable with bounded support. To extend this to ${\rm
Re}(\mu+\lambda)<-2$ we need to regulate the integral. Gelfand identifies a
prescription such that the answer away from integer points (see below) is
defined by analytic continuation in the scaling dimensions from the region
${\rm Re}(\mu+\lambda)>-2$ where the integral converges. This function is
analytic away from $\lambda,\mu=-k-1$ for $k\in\mathbb{Z}_{>}$ where it has
simple poles

${\rm
res}_{\overset{\lambda=-k-1}{\mu=-l-1}}(z^{\lambda}{\bar{z}}^{\mu},\varphi)=\frac{2\pi}{k!l!}\varphi^{(k,l)}(0,0).$
(A.5)

Here the residue is in the single complex variable $s=\mu+\lambda$ with
$n=\mu-\lambda\in\mathbb{Z}$ fixed. Since the difference in scaling dimensions
is integer valued both $\mu$ and $\lambda$ will be integer valued at these
points, and are naturally labeled by $k,l\in\mathbb{Z}_{>}$ as above.

We can formally attach this residue of the integral to the generalized
function $z^{\lambda}{\bar{z}}^{\mu}$

${\rm
res}_{\overset{\lambda=-k-1}{\mu=-l-1}}z^{\lambda}{\bar{z}}^{\mu}=\frac{2\pi}{k!l!}(-1)^{k+l}\delta^{(k,l)}(0,0).$
(A.6)

By introducing a gamma function with the same pole locations we can define an
object whose limits to integral values is the distribution

$\left[\frac{z^{\lambda}{\bar{z}}^{\mu}}{\Gamma\left(\frac{1}{2}(\mu+\lambda)+\frac{1}{2}|\mu-\lambda|+1\right)}\right]_{\overset{\lambda=-k-1}{\mu=-l-1}}=\pi\frac{(-1)^{k+l+j}j!}{k!l!}\delta^{(k,l)}(z,{\bar{z}}),$
(A.7)

where $j=\frac{1}{2}(\mu+\lambda)-\frac{1}{2}|\mu-\lambda|=\min\\{k,l\\}$. In
particular, this implies that

$\partial_{\bar{z}}z^{-1+\alpha}{\bar{z}}^{\alpha}=\alpha
z^{-1+\alpha}{\bar{z}}^{-1+\alpha}~{}~{}\Rightarrow~{}~{}\lim\limits_{\alpha\rightarrow
0}\partial_{\bar{z}}z^{-1+\alpha}{\bar{z}}^{\alpha}=\pi\delta(z,{\bar{z}}).$
(A.8)

This replaces the familiar relation
$\partial_{\bar{z}}z^{-1}=\pi\delta(z,{\bar{z}})$ with a form that we can more
readily analytically continue between signatures. Specifically, we see that
the definitions of these generalized functions are implicitly tied to our
integration contour in (A.2) (here the Riemann sphere).

###  A.2 Representations at Integer Points

Volume 5 of Gelfand [57] discusses infinite dimensional representations of the
Lorentz group. Each such representation is labeled by a pair of complex
numbers $\chi=(n_{1},n_{2})$ where $\left(n_{1}-n_{2}\right)\in\mathbb{Z}$.
The representation operator $T_{\chi}(g)$, for $g\in SL(2,\mathbb{C})$, acts
on the space $D_{\chi}$ of functions $\phi(z,\bar{z})$ via

$T_{\chi}(g)\phi(z,\bar{z})=(\beta
z+\delta)^{n_{1}-1}(\bar{\beta}\bar{z}+\bar{\delta})^{n_{2}-1}\phi\left(\frac{\alpha
z+\gamma}{\beta
z+\delta},\frac{\bar{\alpha}{\bar{z}}+\bar{\gamma}}{\bar{\beta}{\bar{z}}+\bar{\delta}}\right).$
(A.9)

We see that that in terms of our usual notation for the weights $(h,\bar{h})$
we have

$n_{1}=1-2h,\ n_{2}=1-2\bar{h},\ n_{1}-n_{2}=-2(h-\bar{h})=-2J\in\mathbb{Z}.$
(A.10)

The shadow transform induces a simultaneous Weyl reflection on the left and
right handed weights. This corresponds to the map ${\rm
Sh}:D_{\chi}\rightarrow D_{-\chi}$. Namely
$\chi=(n_{1},n_{2})\rightarrow-\chi=(-n_{1},-n_{2})$ while

${\rm Sh}\circ\phi(z,\bar{z})=\int
d^{2}z_{1}(z-z_{1})^{-n_{1}-1}(\bar{z}-\bar{z}_{1})^{-n_{2}-1}\phi(z_{1},\bar{z}_{1}).$
(A.11)

Now suppose $\chi=(h,\bar{h})$ is a representation where $h$ and $\bar{h}$ are
neither simultaneously positive nor simultaneously negative integers. Then the
shadow transform is one-to-one and onto

$\chi=(n_{1},n_{2})\sim-\chi=(-n_{1},-n_{2}).$ (A.12)

The so-called integer points are special representations for which
$\chi=(n_{1},n_{2})$ are either simultaneously positive or simultaneously
negative. These are at the heart of the null state relations and celestial
diamonds of [27, 28, 58, 30]. At integer points the representation $D_{\chi}$
has an invariant subspace and so is reducible. For example suppose $n_{1}$ and
$n_{2}$ are both positive integers. Then $D_{\chi}$ contains polynomials of
the form

$\phi(z,\bar{z})=\sum_{i=0}^{n_{1}-1}\sum_{j=0}^{n_{2}-1}c_{ij}z^{i}\bar{z}^{j},$
(A.13)

which are closed under the action of SL$(2,\mathbb{C})$. This invariant
subspace is called $E_{\chi}$. This has dimension $n_{1}n_{2}$.

Now consider the ‘shadow’ representation $D_{-\chi}$. Within this
representation there is a invariant subspace which can be described as
follows. Consider the space of all functions $\phi(z,\bar{z})\in D_{-\chi}$
which satisfy the conditions

$b_{ij}=\int dzd\bar{z}z^{i}\bar{z}^{j}\phi(z,\bar{z})=0,\ 0\leq i\leq
n_{1}-1,\ 0\leq j\leq n_{2}-1.$ (A.14)

One can show that under the SL$(2,\mathbb{C})$ action on $\phi(z,\bar{z})$ the
numbers $b_{ij}$ transform linearly and so the space described by the
conditions $b_{ij}=0$ is invariant under the action of SL$(2,\mathbb{C})$.
This invariant subspace of $D_{-\chi}$ is denoted by $F_{-\chi}$.

We can phrase this more cleanly as follows. Comparing to our discussion of
generalized functions above we see that the residue of the shadow kernel at
these integer points reduces to a differential operator

$\partial^{n_{1}}\bar{\partial}^{n_{2}}:D_{\chi}\mapsto D_{-\chi}.$ (A.15)

The subspace $E_{\chi}\subset D_{\chi}$ is the kernel of this operator.
Meanwhile $f\in F_{-\chi}$ can be written as
$\partial^{n_{1}}\bar{\partial}^{n_{2}}d$ for some $d\in D_{\chi}$. Namely

$\partial^{n_{1}}\bar{\partial}^{n_{2}}D_{\chi}=F_{-\chi}\subset D_{-\chi}.$
(A.16)

To construct a non-degenerate pairing with elements of $E_{\chi}$ we need to
restrict to the equivalence class $[D_{-\chi}]\sim D_{-\chi}+F_{-\chi}$ which
is the co-kernel of our differential map (A.15). This is closely related to
the equivalence classes encountered in the BMS flux algebra of [37, 38],
however for the leading through sub-subleading radiative soft gravitons only
one of the two (holomorphic or anti-holomorphic) submodules has a primary
descendant.

##  Appendix B Complexifying the Celestial Sphere

We will use this appendix to tie together some loose ends regarding the
contour prescriptions for our modified shadow transform. Our starting point
will be to view the complexified Riemann sphere as the cross section of the
complexified null cone in momentum space. Let $p^{\mu}\in\mathbb{C}^{4}$ be
coordinates on complexified momentum space. The complexified celestial sphere
is then the following quadric in $\mathbb{CP}^{3}$

$-(p^{0})^{2}+(p^{1})^{2}+(p^{2})^{2}+(p^{3})^{2}=0.$ (B.1)

This is known to be bi-holomorphic to $\mathbb{CP}^{1}\times\mathbb{CP}^{1}$
via the Segre embedding
$\mathbb{CP}^{1}\times\mathbb{CP}^{1}\rightarrow\mathbb{CP}^{3}$

$([x:y],[a:b])\mapsto([xa:ya:xb:yb])$ (B.2)

up to a linear transformation. In terms of the standard coordinates for
covering the north pole patch of the celestial sphere

$[x:y]=[1:z],~{}~{}~{}~{}[a:b]=[1:{\bar{z}}]$ (B.3)

the map (B.2) reduces to

$[x^{0}:x^{1}:x^{2}:x^{3}]=[1:z:{\bar{z}}:z{\bar{z}}]=[p^{0}+p^{3},p^{1}+ip^{2},p^{1}-ip^{2}:p^{0}-p^{3}].$
(B.4)

We thus see that the complexified celestial sphere can be naturally though of
as $\mathbb{CP}^{1}\times\mathbb{CP}^{1}$. The celestial sphere in
$\mathbb{R}^{1,3}$ corresponds to the locus $z^{*}={\bar{z}}$, while the locus
$\\{z^{*}=z,{\bar{z}}^{*}={\bar{z}}\\}$ lands us on (a quotient of) the
celestial torus, relevant to $\mathbb{R}^{2,2}$.

As an example of how this complexification of the celestial sphere is useful,
let us examine how a different choice of contour prescription allows [38] to
evade the obstruction encountered by [8]. We wil start with the
$\Delta=3,J=\pm 1$ modes sourcing the subleading soft graviton

$\mathcal{J}=\mathcal{J}_{soft}+\mathcal{J}_{hard},~{}~{}~{}\bar{\mathcal{J}}=\bar{\mathcal{J}}_{soft}+\bar{\mathcal{J}}_{hard},$
(B.5)

where, at linearized order,

$\mathcal{J}_{soft}=-\frac{1}{2}\bar{\partial}^{3}S,~{}~{}~{}\bar{\mathcal{J}}_{soft}=-\frac{1}{2}\partial^{3}\bar{S}.$
(B.6)

The authors of [38] define the celestial stress tensor as follows

${\bf T}(z)=\oint_{C}\frac{d{\bar{z}}}{2\pi
i}\bar{\mathcal{J}}_{soft}(z,{\bar{z}}),~{}~{}~{}\bar{\bf
T}({\bar{z}})=\oint_{C}\frac{dz}{2\pi i}\mathcal{J}_{soft}(z,{\bar{z}}).$
(B.7)

If we formally evaluate the shadow transform (2.3) using the contour
prescription (see also the recent discussion in [59])

$\int d^{2}w\Rightarrow\pi\oint\frac{dw}{2\pi i}\oint\frac{d{\bar{w}}}{2\pi
i}$ (B.8)

of [38] we get $\bar{T}(z,{\bar{z}})\Rightarrow\bar{\bf T}({\bar{z}})$ which
is anti-meromorphic by construction. Now in Appendix A we saw that our
understanding of generalized functions is tied to the parings of the form
(A.2) which involve a choice of metric. For example, a double contour of the
form (B.8) gives

$\lim_{\alpha\rightarrow 0}\alpha\oint\frac{dz}{2\pi
i}z^{-1+\alpha}\oint\frac{d{\bar{z}}}{2\pi i}{\bar{z}}^{-1+\alpha}=0$ (B.9)

in contrast to (A.8). This is consistent with the fact that the Green’s
function for $\partial_{z}^{n_{1}}\partial_{\bar{z}}^{n_{2}}$ is signature-
dependent. For example when we restrict to the codimension 2 locus
$(z,{\bar{z}})\in\mathbb{R}^{1,1}$ rather than
$\partial_{\bar{z}}z^{-1}=\pi\delta(z,{\bar{z}})$ we get
$\partial_{\bar{z}}[\theta({\bar{z}})\delta(z)]=\delta(z)\delta({\bar{z}})$ on
the locus ${\bar{z}}=z^{*}$. The upshot is that (so long as we analytically
continue our dimensions slightly away from the integer points) we can start
from the amplitudes, complexify our mode expansion so that
$(z,{\bar{z}})\in\mathbb{C}^{2}$, take derivatives in $z$ and ${\bar{z}}$ as
we see fit, but when it comes to discussing what contact terms appear our
choice of contour matters.

$w$$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Gamma_{z}$$~{}~{}~{}z$
Figure 1: Branch cut for the shadow kernel in the complexified celestial
sphere coordinate $w$. The blue dots indicate locations of other operator
insertions which from the form of the tree level celestial OPEs are known to
give poles in $w$.

For our discussions in the main text, it’s important to note that the shadow
integral kernel has a branch cut once $z$ and ${\bar{z}}$ are independent.
Indeed, the Green’s function we are using in section 3 can be rephrased in
terms of Cauchy’s residue theorem. Consider a function $f(w)$ that is
holomorphic near the point $w=z$. By Cauchy’s integral formula the $n$th
derivative at $w=z$ can be extracted by the contour integral

$f^{(n)}(z)=\frac{\Gamma(n+1)}{2\pi
i}\oint_{C_{z}}dw\frac{f(w)}{(w-z)^{n+1}},$ (B.10)

where the contour $C_{z}$ is along a closed curve going counterclockwise
around the point $w=z$. If we try to analytically continue $n=1-2h$ away from
an integer value, the integrand develops a branch cut stretching between $w=z$
and $w=\infty$. We can form a closed contour by replacing $C_{z}$ with a
keyhole contour $\Gamma_{z}$. Because $f(w)=\phi(w)\sim w^{-2h}$, the
integrand goes like $w^{-2}$ so there is no residue at infinity. The
contribution from two points on opposite sides of the branch cut in Figure 1
will differ by the following phase

$e^{\pi i(n+1)}-e^{-\pi i(n+1)}=-2i\sin(n\pi)=\frac{2\pi
i}{\Gamma(n+1)\Gamma(-n)}$ (B.11)

so that

$f^{(n)}(z)\ni\frac{1}{\Gamma(-n)}\int_{-\infty}^{z}\frac{f(w)}{(w-z)^{n+1}}$
(B.12)

matching (3.7). The small arc and branch cut contributions are related by the
residues that appear from collinear limits with other operator insertions in a
given $\cal S$-matrix element (indicated schematically by the blue dots in
figure 1).

##  Appendix C OPE between modified stress tensors:
$\bar{T}_{mod}\bar{T}_{mod}$

Let’s start by reviewing the elements of the derivation in [8] that get
modified once we take into account the reparameterization mode. In order to
reproduce the standard centerless Virasoro symmetry, a certain smearing of the
subleading soft mode must be shown to vanish

$\begin{gathered}\langle{\bar{T}(z,{\bar{z}})\bar{T}(w,{\bar{w}})\prod_{i=2}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle=\text{Standard
terms with $c=0$ }~{}~{}~{}~{}~{}\\\
~{}~{}~{}~{}+\frac{6}{\pi(\bar{w}-\bar{z})^{2}}\int
d^{2}z_{1}\frac{1}{(\bar{z}-\bar{z}_{1})^{2}(\bar{w}-\bar{z}_{1})^{2}}\langle{S(z_{1},\bar{z}_{1})\prod_{i=2}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle.\end{gathered}$
(C.1)

One needs to be careful, since while the angular dependence of the regulated
$z_{1}$ integral vanishes due to conformal invariance (c.f. section 4.1 of
[8]) the in the regulated $z_{1}$, there is an overall factor of $\Gamma[0]$.
One thus needs to be careful not to drop subleading terms in the
$\Delta\rightarrow 0$ expansion of the object it is multiplying. By taking a
$w$ derivative of both sides, we can see that there is an obstruction to this
integral vanishing via (2.5) and (LABEL:Sdec).

We can now show that if we repeat the same manipulations that led to the
obstruction (LABEL:TdecT) with $\bar{T}$ replaced with $\bar{T}_{\rm mod}$,
the non-contact terms vanish. The correlation function we are interested in
takes the form

$\displaystyle\begin{aligned}
&\langle{\bar{T}_{mod}(z,\bar{z})\bar{T}_{mod}(w,\bar{w})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle=\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\langle{\bar{T}(z,\bar{z})\bar{T}(w,\bar{w})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle+\langle{\frac{1}{2}\bar{\partial}^{3}\epsilon_{S}(z,\bar{z})\frac{1}{2}\bar{\partial}^{3}\epsilon_{S}(w,\bar{w})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\langle{\bar{T}(z,\bar{z})\frac{1}{2}\bar{\partial}^{3}\epsilon_{S}(w,\bar{w})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle+\langle{\frac{1}{2}\bar{\partial}^{3}\epsilon_{S}(z,\bar{z})\bar{T}(w,\bar{w})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle.\end{aligned}$

(C.2)

Now, as discussed in section 2, the first term is (C.1). Using that

$\langle{\bar{\partial}^{3}\epsilon_{S}(z,\bar{z})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle=0$
(C.3)

as an operator equation, and that we can choose the memory operators (2.19) to
have vanishing correlators [30]

$\langle\frac{1}{2}\bar{\partial}^{3}\epsilon_{S}(z,\bar{z})\frac{1}{2}\bar{\partial}^{3}\epsilon_{S}(w,\bar{w})\rangle=0$
(C.4)

we see that the second term on the right hand side of (C.2) vanishes. To
obtain the mixed correlators involving $\bar{T}$ and $\epsilon_{S}$ we use the
relation

$\displaystyle\begin{gathered}\langle{\bar{T}(z,\bar{z})\bar{\partial}^{3}S(w,\bar{w})\prod_{i=1}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle=\left(-\frac{24}{\left(\bar{z}-\bar{w}\right)^{5}}-\frac{12}{\left(\bar{z}-\bar{w}\right)^{4}}\frac{\partial}{\partial\bar{w}}\right)\langle{S(w,\bar{w})\prod_{i=1}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle\end{gathered}$

(C.5)

and the fact that $S(z,\bar{z})$ and $\epsilon_{S}(z,\bar{z})$ transform in
the same way under $\overline{Vir}$ transformations. This follows from the
definition $S=\partial\epsilon_{S}$ and leads to

$\displaystyle\begin{gathered}\langle{\bar{T}(z,\bar{z})\frac{1}{2}\bar{\partial}^{3}\epsilon_{S}(w,\bar{w})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle=\left(-\frac{12}{\left(\bar{z}-\bar{w}\right)^{5}}-\frac{6}{\left(\bar{z}-\bar{w}\right)^{4}}\frac{\partial}{\partial\bar{w}}\right)\langle{\epsilon_{S}(w,\bar{w})\prod_{i=1}^{n}\phi_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\rangle,\end{gathered}$

(C.6)

where we have also used (C.3). Putting everything together we get the standard
answer with zero central charge

$\begin{gathered}\langle\bar{T}_{mod}(z,\bar{z})\bar{T}_{mod}(w,\bar{w})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})\rangle=\\\
\left(\frac{2}{(\bar{z}-\bar{w})^{2}}+\frac{1}{\bar{z}-\bar{w}}\frac{\partial}{\partial\bar{w}}\right)\langle{\bar{T}_{mod}(w,\bar{w})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle\\\
+\sum_{i}\left(\frac{\bar{h}_{i}}{(\bar{z}-\bar{z}_{i})^{2}}+\frac{1}{\bar{z}-\bar{z}_{i}}\frac{\partial}{\partial\bar{z}_{i}}\right)\langle{\bar{T}_{mod}(w,\bar{w})\prod_{i}\phi_{i}(z_{i},\bar{z}_{i})}\rangle.\end{gathered}$
(C.7)

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