aainstitutetext: Indian Institute of Science Education and Research Bhopal,
Bhopal Bypass, Bhopal 462066, India.bbinstitutetext: NHETC and Department of
physics and Astronomy, Rutgers University, 126 Frelinghuysen Rd., piscataway
NJ 08855, USA

# Asymptotic Symmetry algebra of $\mathcal{N}=8$ Supergravity

Nabamita Banerjee a Tabasum Rahnuma b and Ranveer Kumar Singh
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The asymptotic symmetry algebra of $\mathcal{N}=1$ supergravity was recently
constructed using the well-known $2$D celestial CFT (CCFT) technique
Fotopoulos:2020bqj . In this paper, we extend the construction to the
maximally supersymmetric four dimensional $\mathcal{N}=8$ supergravity theory
in asymptotically flat spacetime and construct the extended asymptotic
symmetry algebra, which we call $\mathcal{N}=8$ $\mathfrak{sbms}_{4}$. We use
the celestial CFT technique to find the appropriate currents for extensions of
$\mathcal{N}=8$ super-Poincaré and $\mathrm{SU}(8)_{R}$ R-symmetry current
algebra on the celestial sphere $\mathcal{CS}^{2}$. We generalise the
definition of shadow transformations and show that there is no infinite
dimensional extension of the global $\mathrm{SU}(8)_{R}$ algebra in the
theory.

##  1 Introduction

The physical observables of a theory are encoded in the symmetries of that
theory. This makes the study of symmetries very important. Furthermore, it has
been observed that in gauge and gravity theories, there is an enhancement of
symmetry at the boundaries. For asymptotic boundaries, such enhanced
symmetries are known as the asymptotic symmetries. In four dimensions these
asymptotic symmetries have been studied for both gauge and gravity theories
including the $\mathcal{N}=1$ supergravity He:2014cra ; He:2015zea ;
Dumitrescu:2015fej ; PhysRevLett.105.111103 ; PhysRev.128.2851 ;
Strominger:2013lka ; Fotopoulos:2020bqj . These symmetries in case of gravity
and gauge theories are popularly known as BMS (Bondi-Metzner-Sachs) and large
gauge symmetries respectively Bondi:1962px ; Sachs:1962wk ; Sachs:1962zza ;
Strominger:2017zoo ; He:2014laa ; Barnich:2011mi ; Barnich:2013axa ;
Strominger:2013jfa ; Gabai:2016kuf . These infinite dimensional asymptotic
symmetries also have experimental implications as in gauge and gravitaional
memory effects which are classical observables Favata:2010zu ;
PhysRevLett.67.1486 ; Zeldovich:1974gvh ; Susskind:2015hpa ; Pasterski:2015zua
. Moreover, a deeper understanding of these symmetries might help in
understanding the black hole microscopics Carlip:2017xne ; Sen:2007qy ;
Mandal:2010cj ; Dabholkar:2004yr ; Carlip:2012ff . This necessitates the
computation and analysis of asymptotic symmetries.

Another implication of the asymptotic symmetries is the soft theorems. It has
been shown that soft theorems are the Ward identities of the asymptotic
symmetries Strominger:2013jfa ; He:2014laa ; Campiglia:2019wxe . Let us
discuss this relation in a bit detail. In general every symmetry leads to
constraints on physical observable such as the scattering amplitudes. Such
constraints are known as Ward identities. For the Ward identities of
asymptotic symmetries we need to consider the amplitude in the soft limit. The
soft limit of an amplitude is defined by taking the momenta of one or more
external particles to zero. Quite generally, under soft limit, the amplitude
factorises into a universal (soft) factor which contains the divergent part of
the amplitude times the amplitude without the soft particle(s) insertions.
This factorization is known as soft theorem. In other words, soft theorems are
the Ward identities of asymptotic symmetries. The BMS and large gauge
symmetries lead to soft graviton and soft photon theorem Strominger:2017zoo .

Another important limit of amplitudes is the collinear limit in which the
momenta of two external particles are taken to be collinear. Again the
amplitude factorises into a collinear factor containing the divergence times
the amplitude with the collinear particles replaced by another particle
Taylor:2017sph . The collinear limit of amplitude turns into an operator
product expansion (OPE) of conformal operators of the Celestial Conformal
Field theory (CCFT) Pasterski:2021raf ; Pasterski:2021rjz ; Donnay:2020guq ;
Fan:2019emx  as identical momentum directions corresponds to the same operator
insertion points on the celestial sphere111The celestial sphere is the Riemann
sphere on the boundary of the Minskowski space. (which we denote by
$\mathcal{CS}^{2}$). An interesting fact is that the soft and collinear limits
of scattering amplitudes can be used to read off the asymptotic symmetries in
the context of CCFT Donnay:2020guq ; Fotopoulos:2019vac . It turns out that to
calculate the asymptotic symmetries of a theory, we need to probe the
universal soft and collinear sectors of the scattering amplitudes. This has
been used to reproduce the BMS algebra in Fotopoulos:2019vac  and
Banerjee:2022wht  for pure gravity and large gauge algebra for Einstein Yang-
Mills theory Banerjee:2021uxe . Recently, it has also been used to compute the
$\mathcal{N}=1$ supersymmetric extension of the BMS algebra Fotopoulos:2020bqj
.

In this paper, we calculate the asymptotic symmetries of the four dimensional
maximally supersymmetric $\mathcal{N}=8$ supergravity using the CCFT
prescription. In celestial CFT of supergravity, the stress tensor is generated
by the shadow transform of the soft graviton operator suitably modified to
obtain the correct OPE Banerjee:2022wht 222see Section 3 for more details,
while the supercurrent is generated by the soft gravitino operator
Fotopoulos:2020bqj . For $\mathcal{N}>1$, the global symmetry algebra contains
an additional R-symmetry and hence naïvely one would expect that the
asymptotic algebra would contain an infinite dimensional extension of the
global R-symmetry algebra as well. It was shown in Banerjee:2022abf  that for
$\mathcal{N}=2$, even for the $\mathrm{U}(1)^{\mathcal{N}}$ subgroup of the
R-symmetry group $\mathrm{U}(\mathcal{N})$ which only scales the supercharges,
such an infinite dimensional extension is mathematically inconsistent. For the
present paper, we study the celestial amplitudes of $\mathcal{N}=8$
supergravity and use the soft and collinear limits calculated in a companion
paper tab  to compute the Ward identities and the OPE of conformal operators
in the corresponding CCFT. We then construct the stress tensor and the
supercurrents of the theory using the shadow transforms of soft graviton and
soft gravitino operators. Since the scalars and graviphotinos do not have soft
divergences (see tab ), we are only left with soft graviphoton operators. The
R-symmetry current (if any) can then only be constructed using the soft
graviphoton operators. We construct the most general such operator present in
the CCFT and show that the operator is trivial by requiring that the modes of
this operator extend the $\mathrm{SU}(8)_{R}$ R-symmetry algebra.

The paper is organised as follows: in Section 2 we set up our notations and
record some definitions and results about the soft and collinear limits in the
CCFT of $\mathcal{N}=8$ supergravity used later in the paper. In Section 3 we
construct the symmetry currents and compute their OPEs. We also construct the
possible R-symmetry currents and show that the requirements of R-symmetry
extension make the current trivial. Finally in Section 4 we list the full
$\mathcal{N}=8\leavevmode\nobreak\ \mathfrak{sbms}_{4}$ algebra. We conclude
in Section 5 by summarising our results and emphasizing our future goals of
the study. The Appendices contain the OPEs of various conformal operators in
the Mellin basis computed from the results in tab  and a detailed calculation
of the OPE of the possible R-symmetry currents.

##  2 Notations and preliminaries

In this section, we set up the notations for celestial amplitudes and soft and
collinear limits in supergravity.

###  2.1 Celestial amplitudes

Recall that helicity spinors are left and right handed representations of the
Lorentz group $\mathrm{SO}(1,3)\sim\mathrm{SL}(2,\mathbb{C})$. We denote the
left and right handed helicity spinors by $\lambda_{\alpha}$ and
$\tilde{\lambda}^{\dot{\alpha}}$ respectively. A given null momentum $p^{\mu}$
can be written as a bispinor

$p^{\alpha\dot{\alpha}}=\sigma_{\mu}^{\alpha\dot{\alpha}}p^{\mu}=\left(\begin{array}[]{cc}p^{0}+p^{3}&p^{1}-ip^{2}\\\
p^{1}+ip^{2}&p^{0}-p^{3}\end{array}\right)=\lambda^{\alpha}\tilde{\lambda}^{\dot{\alpha}}$
(1)

where
$\sigma_{\mu}^{\alpha\dot{\alpha}}=(1,\sigma_{x},\sigma_{y},\sigma_{z})$. For
real physical momentum, the two spinors are related by complex conjugation
$(\tilde{\lambda}^{\dot{\alpha}})^{*}=\lambda_{\alpha}$.

We now want to study scattering kinematics on the celestial sphere. We use the
Bondi coordinates $(u,r,z,\bar{z})$ on the Minkowski space where $(z,\bar{z})$
parametrises the celestial sphere $\mathcal{CS}^{2}$ at null infinity. The
Lorentz group $\mathrm{SL}(2,\mathbb{C})$ acts on $\mathcal{CS}^{2}$ as
follows:

$(z,\bar{z})\longmapsto\left(\frac{az+b}{cz+d},\frac{\bar{a}\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}}\right),\quad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{C}).$

A general null momentum vector $p^{\mu}$ can be parametrized as

$p^{\mu}=\omega q^{\mu},\quad
q^{\mu}=\frac{1}{2}\left(1+|z|^{2},z+\bar{z},-i(z-\bar{z}),1-|z|^{2}\right),$

where $q^{\mu}$ is a null vector, $\omega$ is identified with the light cone
energy and all the particles momenta are taking to be outgoing. Under the
Lorentz group the four momentum transforms as a Lorentz vector
$p^{\mu}\mapsto\Lambda^{\mu}_{\leavevmode\nobreak\ \nu}p^{\nu}$. This induces
the following transformation of $w$ and $q^{\mu}$ as

$\omega\mapsto(cz+d)(\bar{c}\bar{z}+\bar{d})\omega,\quad q^{\mu}\mapsto
q^{\prime\mu}=(cz+d)^{-1}(\bar{c}\bar{z}+\bar{d})^{-1}\Lambda_{\leavevmode\nobreak\
\nu}^{\mu}q^{\nu}.$

In the bispinor notation, we can write the basic null momentum vector
$q^{\mu}$ as

$q^{\alpha\dot{\alpha}}=\sigma_{\mu}^{\alpha\dot{\alpha}}q^{\mu}=\left(\begin{array}[]{cc}1&\;\bar{z}\\\
z&\;z\bar{z}\end{array}\right)=\left(\begin{array}[]{l}1\\\
z\end{array}\right)\left(\begin{array}[]{ll}1&\bar{z}\end{array}\right),$ (2)

where ${\sigma}^{\mu}\equiv(1,\sigma_{x},\sigma_{y},\sigma_{z})$ are two
dimensional identity and Pauli matrices. Further introducing the angle and
square notation for the left and right handed momentum spinors, we have:

$\lambda^{\alpha}\equiv\langle
p|^{\alpha}=\sqrt{\omega}\left(\begin{array}[]{l}1\\\
z\end{array}\right)=\sqrt{\omega}\langle
q|^{\alpha},\quad\tilde{\lambda}^{\dot{\alpha}}\equiv|p]^{\dot{\alpha}}=\sqrt{\omega}\left(\begin{array}[]{l}1\\\
\bar{z}\end{array}\right)=\sqrt{\omega}|q]^{\dot{\alpha}},$ (3)

where we we write

$\langle q|^{\alpha}=\left(\begin{array}[]{l}1\\\
z\end{array}\right),\quad|q]^{\dot{\alpha}}=\left(\begin{array}[]{c}1\\\
\bar{z}\end{array}\right).$ (4)

The inner product of momenta can then be written in terms of the angle and
square brackets of the corresponding spinors which are now given by

$\langle
ij\rangle=-\sqrt{\omega_{i}\omega_{j}}z_{ij},\quad[ij]=\sqrt{\omega_{i}\omega_{j}}\bar{z}_{ij},$
(5)

where $z_{ij}=z_{i}-z_{j},\bar{z}_{ij}=\bar{z}_{i}-\bar{z}_{j}$.

We can now Mellin transform the fields in the bulk to get conformal primaries
on the celestial sphere. The massless conformal primary of conformal dimension
$\Delta$ are given by

$\varphi_{\Delta}^{\pm}\left(X^{\mu},z,\bar{z}\right)=\int_{0}^{\infty}d\omega\omega^{\Delta-1}e^{\pm
i\omega q\cdot X-\epsilon\omega}=\frac{(\mp
i)^{\Delta}\Gamma(\Delta)}{(-q\cdot X\mp i\epsilon)^{\Delta}}.$ (6)

The conformal primaries for nontrivial spins are then given by
Fotopoulos:2020bqj ,

$\begin{split}&\psi_{\Delta,\ell=-1/2;\alpha}^{\pm}(X,z,\bar{z})=|q\rangle_{\alpha}\varphi^{\pm}_{\Delta+\frac{1}{2}}(X,z,\bar{z})\\\
&\psi_{\Delta,\ell=1/2}^{\pm;\dot{\alpha}}(X,z,\bar{z})=|q]^{\dot{\alpha}}\varphi^{\pm}_{\Delta+\frac{1}{2}}(X,z,\bar{z})\\\
&V^{\mu\pm}_{\Delta,\ell=\pm 1}(X,z,\bar{z})=\epsilon^{\mu}_{\ell=\pm
1}(q,r)\varphi^{\pm}_{\Delta}(X,z,\bar{z})\\\ &H_{\Delta,\ell=\pm
2}^{\mu\nu\pm}(X,z,\bar{z})=\epsilon_{\ell=\pm 1}^{\mu}(q,r)V_{\Delta,\ell=\pm
1}^{\nu\pm}(X,z,\bar{z})\\\
&\psi_{\Delta,\ell=-3/2}^{\mu\pm}(X,z,\bar{z})=\epsilon_{\ell=-1}^{\mu}(q,r)\psi_{\Delta,\ell=-1/2}^{\pm}(X,z,\bar{z})\\\
&\bar{\psi}_{\Delta,\ell=+3/2}^{\mu\pm}(X,z,\bar{z})=\epsilon_{\ell=+1}^{\mu}(q,r)\bar{\psi}_{\Delta,\ell=+1/2}^{\pm}(X,z,\bar{z})\end{split}$
(7)

where the polarisations are given by

$\epsilon_{\ell=+1}^{\mu}(q,r)=\frac{\left\langle
r\left|\sigma^{\mu}\right|q\right]}{\sqrt{2}\langle
rq\rangle},\quad\epsilon_{\ell=-1}^{\mu}(q,r)=\frac{\left[r\left|\bar{\sigma}^{\mu}\right|q\right\rangle}{\sqrt{2}[qr]}$
(8)

with $r$ is a reference null vector and
$\bar{\sigma}^{\mu}\equiv(1,-\sigma_{x},-\sigma_{y},-\sigma_{z})$. One can
further define inner product on these conformal wavepackets Fotopoulos:2020bqj
. These conformal wavepackets are normalisable only when the conformal
dimension $\Delta$ belongs to the principle continuous series, that is
$\Delta=1+i\lambda$ with $\lambda\in\mathbb{R}$. In a scattering process, we
take all momenta to be outgoing. We now define celestial amplitude or
celestial correlator on $\mathcal{CS}^{2}$ as the Mellin transform of the
amplitudes:

$\left\langle\prod_{n=1}^{N}\mathcal{O}_{\Delta_{n},\ell_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle\equiv\left(\prod_{n=1}^{N}\int
d\omega_{n}\omega_{n}^{\Delta_{n}-1}\right)\delta^{(4)}\left(\sum_{n=1}^{N}\omega_{n}q_{n}\right)A_{\ell_{1}\ldots\ell_{N}}\left(\omega_{n},z_{n},\bar{z}_{n}\right),$
(9)

where $A_{\ell_{1},\dots,\ell_{n}}$ is the bulk amplitude with external
particles with helicities $\ell_{1},\dots,\ell_{n}$. The celestial correlators
can be shown to transform as a conformal correlator under
$\mathrm{SL}(2,\mathbb{C})$:

$\left\langle\prod_{n=1}^{N}\mathcal{O}_{\Delta_{n},\ell_{n}}\left(\frac{az_{n}+b}{cz_{n}+d},\frac{\bar{a}\bar{z}_{n}+\bar{b}}{\bar{c}\bar{z}_{n}+\bar{d}}\right)\right\rangle=\prod_{i=1}^{N}(cz_{i}+d)^{\Delta_{i}+\ell_{i}}(\bar{c}\bar{z}_{i}+\bar{d})^{\Delta_{i}-\ell_{i}}\left\langle\prod_{n=1}^{N}\mathcal{O}_{\Delta_{n},\ell_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle.$
(10)

where

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{C}).$ (11)

###  2.2 OPEs of celestial operators in $\mathcal{N}=8$ Supergravity

Let $\\{\eta_{A}\\}_{A=1}^{8}$ be the Grassmann coordinates on the
$\mathcal{N}=8$ superspace. We can package the on-shell degrees of freedom in
$\mathcal{N}=8$ supergravity in an on-shell superfield defined as

$\begin{split}\Psi(p,\eta)&=H^{+}(p)+\eta_{A}\psi^{A}_{+}(p)+\eta_{AB}G^{AB}_{+}(p)+\eta_{ABC}\chi^{ABC}_{+}(p)\\\
&+\eta_{ABCD}\Phi^{ABCD}(p)+\tilde{\eta}^{ABC}\chi_{ABC}^{-}(p)+\tilde{\eta}^{AB}G_{AB}^{-}(p)+\tilde{\eta}^{A}\psi_{A}^{-}(p)+\tilde{\eta}H^{-}(p)\end{split}$
(12)

where we have introduced the notation

$\begin{split}&\eta_{A_{1}\ldots
A_{n}}\equiv\frac{1}{n!}\eta_{A_{1}}\ldots\eta_{A_{2}}\\\
&\tilde{\eta}^{A_{1}\ldots A_{n}}\equiv\epsilon^{A_{1}\ldots A_{n}B_{1}\ldots
B_{8-n}}\eta_{B^{1}\ldots B^{8-n}}\\\
&\tilde{\eta}\equiv\prod_{A=1}^{8}\eta^{A}.\end{split}$ (13)

The fields $H^{\pm}$ represent positive and negative helicity graviton,
$G^{AB}_{+}$ and $G_{AB}^{-}$ represent positive and negative helicity
graviphotons, $\psi^{A}_{+}$ and $\psi_{A}^{-}$ represent positive and
negative helicity gravitinos, $\chi^{ABC}_{+}$ and $\chi_{ABC}^{-}$ represent
positive and negative helicity graviphotinos and finally $\Phi^{ABCD}$
represent the real scalars. The superamplitude is then defined by

$\mathcal{M}_{n}(\\{p_{1},\eta^{1}\\},\dots\\{p_{n},\eta^{n}\\})=\langle\Psi_{1}(p_{1},\eta^{1})\dots\Psi_{n}(p_{n},\eta^{n})\rangle.$
(14)

This superfield can be Mellin transformed in the usual way to obtain a
celestial superfield on $\mathcal{CS}^{2}$, but it turns out that the
component fields will have same conformal dimension Jiang:2021xzy . This is
not appropriate to work with since we want the component fields to have
conformal dimensions according to their spin. Thus we work with the so called
quasi-on-shell superfield Jiang:2021xzy  defined as

$\begin{split}\Psi_{\Delta}(z,\bar{z},\eta)&=H^{+}_{\Delta}(z,\bar{z})+\eta_{A}\psi_{\Delta}^{A}(z,\bar{z})+\eta_{AB}G^{AB}_{\Delta}(z,\bar{z})+\eta_{ABC}\chi^{ABC}_{\Delta}(z,\bar{z})\\\
&+\eta_{ABCD}\Phi^{ABCD}_{\Delta}(z,\bar{z})+\tilde{\eta}^{ABC}\bar{\chi}_{ABC\;\Delta}(z,\bar{z})+\tilde{\eta}^{AB}\bar{G}_{AB\;\Delta}(z,\bar{z})\\\
&+\tilde{\eta}^{A}\bar{\psi}_{A\;\Delta}(z,\bar{z})+\tilde{\eta}H^{-}_{\Delta}(z,\bar{z}),\end{split}$
(15)

where the components are the Mellin transforms of the components fields of
$\Psi(p,n)$, all with scaling dimension $\Delta$ as defined in (7). The
celestial correlator for the component fields can then be defined as in (9).
Using the collinear limit of the bulk amplitude, the OPEs of the celestial
operators can be computed. To do this computation, we use the collinear limits
computed in tab . As an example, we calculate the OPE of two graviton
operator. The celestial correlator is given by,

$\begin{split}\langle\mathcal{O}_{\Delta_{1},+2}\mathcal{O}_{\Delta_{2},+2}\dots\mathcal{O}_{\Delta_{n},\ell_{n}}\rangle&=\left(\prod_{j=1}^{n}\int_{0}^{\infty}d\omega_{j}\omega_{j}^{\Delta_{j}-1}\right)\delta^{4}\left(\sum_{i}\omega_{i}q_{i}\right)M_{n}\left(1^{+2},2^{+2},\dots,n\right)\\\
&=\left(\prod_{j=3}^{n}\int_{0}^{\infty}d\omega_{j}\omega_{j}^{\Delta_{j}-1}\int_{0}^{\infty}d\omega_{1}\int_{0}^{\infty}d\omega_{2}\omega_{1}^{\Delta_{1}-1}\omega_{2}^{\Delta_{2}-1}\right)\\\
&\qquad\times\delta^{4}\left(\sum_{i=3}^{n}\omega_{i}q_{i}+\omega_{p}q_{p}\right)\frac{\omega_{p}^{2}}{\omega_{1}\omega_{2}}\frac{\bar{z}_{12}}{z_{12}}M_{n-1}\left(p^{+2},\ldots,n\right)\end{split}$
(16)

where $M_{n}$ is the bulk amplitude of component fields and we used the
collinear limit

$M_{n}\left(1^{+2},2^{+2},\dots,n\right)=\frac{\omega_{p}^{2}}{\omega_{1}\omega_{2}}\frac{\bar{z}_{12}}{z_{12}}M_{n-1}\left(p^{+2},\ldots,n\right).$
(17)

Here $p_{i}=\omega_{i}q_{i},\leavevmode\nobreak\ i=1,2$, the momenta along the
collinear channel is $p=p_{1}+p_{2}=\omega_{p}q_{p}$ with
$\omega_{p}=\omega_{1}+\omega_{2}$. Now we use the following integral
Jiang:2021xzy :

$\int_{0}^{\infty}d\omega_{2}\;\omega_{2}^{\Delta_{2}-1}\int_{0}^{\infty}d\omega_{1}\;\omega_{1}^{\Delta_{1}-1}\;\omega_{1}^{\alpha}\;\omega_{2}^{\beta}\;\omega_{p}^{\gamma}\;f\left(\omega_{p}\right)=B\left(\Delta_{1}+\alpha,\Delta_{2}+\beta\right)\int_{0}^{\infty}d\omega_{p}\;\omega_{p}^{\Delta_{p}-1}f(\omega_{p})$
(18)

where $\omega_{p}=\omega_{1}+\omega_{2}$ and
$\Delta_{p}=\Delta_{1}+\Delta_{2}+\alpha+\beta+\gamma$ and

$B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ (19)

is the Euler beta function. We get

$\begin{split}\langle\mathcal{O}_{\Delta_{1},+2}\mathcal{O}_{\Delta_{2},+2}\dots\mathcal{O}_{\Delta_{n},\ell_{n}}\rangle&=\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-1,\Delta_{2}-1\right)\left(\prod_{j=3}^{n}\int_{0}^{\infty}d\omega_{j}\omega_{j}^{\Delta_{j}-1}\int_{0}^{\infty}d\omega_{p}\omega_{p}^{\Delta_{1}+\Delta_{2}-1}\right)\;\\\
&\times\delta^{4}\left(\sum_{i=3}^{n}\omega_{i}q_{i}+\omega_{p}q_{p}\right)M_{n-1}\left(p^{+2},3,\ldots,n\right)\\\
&=\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-1,\Delta_{2}-1\right)\;\langle\mathcal{O}_{\Delta_{1}+\Delta_{2},+2}\mathcal{O}_{\Delta_{3},\ell_{3}}\dots\mathcal{O}_{\Delta_{n},\ell_{n}}\rangle\end{split}$
(20)

This gives the OPE corresponding to the two positive helicity graviton
operators,

$\mathcal{O}_{\Delta_{1},+2}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{\Delta_{2},+2}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-1,\Delta_{2}-1\right)\mathcal{O}_{\Delta_{1}+\Delta_{2},+2}\left(z_{2},\bar{z}_{2}\right)$
(21)

Similarly for negative helicity gluon we have the collinear amplitude,

$\begin{split}M_{n}\left(1^{-2},2^{-2},\ldots\right)&=\frac{\omega_{p}^{2}}{\omega_{1}\omega_{2}}\frac{z_{12}}{\bar{z}_{12}}M_{n-1}\left(p^{-2},\ldots,n\right)\end{split}$
(22)

Hence the OPE

$\begin{split}\mathcal{O}_{\Delta_{1},-2}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{\Delta_{2},-2}\left(z_{2},\bar{z}_{2}\right)\sim\frac{z_{12}}{\bar{z}_{12}}B\left(\Delta_{1}-1,\Delta_{2}-1\right)\mathcal{O}_{\Delta_{1}+\Delta_{2},+2}\left(z_{2},\bar{z}_{2}\right)\end{split}$
(23)

The collinear limit of two opposite helicity gravitons is

$\begin{split}M_{n}\left(1^{+2},2^{-2},3,\ldots,n\right)=&\frac{\omega_{1}^{3}}{\omega_{p}^{2}\omega_{2}}\frac{\bar{z}_{12}}{z_{12}}M_{n-1}\left(p^{-2},3,\ldots,n\right)\\\
&+\frac{\omega_{2}^{3}}{\omega_{p}^{2}\omega_{1}}\frac{z_{12}}{\bar{z}_{12}}M_{n-1}\left(p^{+2},3,\ldots,n\right)\end{split}$
(24)

which gives us the OPE

$\begin{split}\mathcal{O}_{\Delta_{1},+2}\left(z_{1},z_{1}\right)\mathcal{O}_{\Delta_{2},-2}\left(z_{2},\bar{z}_{2}\right)&=B\left(\Delta_{1}+3,\Delta_{2}-1\right)\frac{\bar{z}_{12}}{z_{12}}\mathcal{O}_{\Delta_{1}+\Delta_{2},-2}\left(z_{2},\bar{z}_{2}\right)\\\
&+B\left(\Delta_{1}-1,\Delta_{2}+3\right)\frac{z_{12}}{\bar{z}_{12}}\mathcal{O}_{\Delta_{1}+\Delta_{2},+2}\left(z_{2},\bar{z}_{2}\right)\\\
\end{split}$ (25)

One can calculate the OPEs of all other component fields in a similar way
using the collinear limits. The results are listed in Appendix A.

###  2.3 Soft operators in $\mathcal{N}=8$ supergravity

In the last section we discussed about the collinear limits of amplitudes. In
this section we are looking at their soft limits. As we know, a soft momenta
$p\to 0$ can be written as $\omega_{p}\to 0$ on the celestial sphere and hence
an amplitude written in the celestial coordinates can be analysed in the soft
limit of any of the external momenta. The result is a soft theorem which
expresses an $n$-point amplitude with a soft external momenta $p$ in terms of
an $(n-1)$-point amplitude along with a soft factor given by powers of
$\omega_{p}^{-1}$. The various powers of $\omega_{p}^{-1}$ then correspond to
leading, sub-leading, sub-sub-leading soft theorems and so on. Let us first
define the celestial superamplitude as the Mellin transform of superamplitude:

$\begin{split}\left\langle\prod_{n=1}^{N}\mathcal{O}_{\Delta_{n}}\left(z_{n},\bar{z}_{n},\eta^{n}\right)\right\rangle\equiv\left(\prod_{n=1}^{N}\int
d\omega_{n}\omega_{n}^{\Delta_{n}-1}\right)\delta^{(4)}\left(\sum_{n=1}^{N}\omega_{n}q_{n}\right)\times\\\
\mathcal{M}_{N}\left(\\{\omega_{1},z_{1},\bar{z}_{1},\eta^{1}\\},\dots\\{\omega_{N},z_{N},\bar{z}_{N},\eta^{N}\\}\right),\end{split}$
(26)

where
$\mathcal{M}_{N}\left(\\{\omega_{1},z_{1},\bar{z}_{1},\eta^{1}\\},\dots\\{\omega_{N},z_{N},\bar{z}_{N},\eta^{N}\\}\right)$
is the superampltude (14) written in the celestial basis. We also denote it
simply by $\mathcal{M}_{N}\left(1,2,\dots,N\right)$. One can then expand both
sides of (26) in the Grassmann parameter $\eta_{i}$ and compare the
coefficients to get the celestial amplitude of various component fields. The
above expression is identical to that of (9), with the explicit incorporation
of the Grassmann factors in the scattering amplitudes.

####  2.3.1 Soft graviton operators

The leading and sub-leading soft factors corresponding to $\omega_{p}^{-1}$
and $\omega_{p}^{0}$ were calculated in tab  using double copy relations. Here
we only present relevant results and refer the readers to tab  for further
details. In the celestial basis, the leading soft factor is given by

$\begin{split}\mathcal{M}_{N}(\cdots,j-1,j,j+1,\cdots)\xrightarrow{\omega_{j}\rightarrow
0}\frac{1}{\omega_{j}}\sum_{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}^{N}\omega_{i}\;z_{ji}\;\bar{z}_{ji}\left[\frac{z^{2}_{j-1,i}}{z^{2}_{j-1,j}\;z_{j,i}^{2}}+\frac{\bar{z}_{j-1,i}^{2}\;}{\bar{z}_{j-1}^{2}\;\bar{z}_{j,i}^{2}}\delta^{4}\left(\eta^{j}\right)\right]\\\
\times\mathcal{M}_{N-1}(\cdots,j-1,j+1,\cdots)\end{split}$ (27)

One can now get the soft limit in terms of celestial superamplitude in a
straightforward way. We have

$\displaystyle\left\langle\prod_{n=1}^{N}\lim_{\Delta_{j}\rightarrow
1}(\Delta_{j}-1)\mathcal{O}_{\Delta_{n}}\left(z_{n},\bar{z}_{n},\eta^{n}\right)\right\rangle=\left(\prod_{\begin{subarray}{c}n=1\\\
n\neq j\end{subarray}}^{N}\int
d\omega_{n}\;\omega_{n}^{\Delta_{n}-1}\right)\lim_{\Delta_{j}\to
1}\int_{0}^{\infty}d\omega_{j}(\Delta_{j}-1)\omega_{j}^{\Delta_{j}-1}$ (28)
$\displaystyle\hskip
142.26378pt\times\delta^{(4)}\left(\sum_{\begin{subarray}{c}k=1\\\ k\neq
j\end{subarray}}^{N}\omega_{k}q_{k}\right)\mathcal{M}_{N}\left(1,\dots,n,\dots,N\right)$
$\displaystyle=\left(\prod_{\begin{subarray}{c}n=1\\\ n\neq
j\end{subarray}}^{N}\int
d\omega_{n}\;\omega_{n}^{\Delta_{n}-1}\right)\int_{0}^{\infty}d\omega_{j}\frac{d}{d\omega_{j}}\left(\lim_{\Delta_{j}\to
1}\omega_{j}^{\Delta_{j}-1}\right)\delta^{(4)}\left(\sum_{\begin{subarray}{c}k=1\\\
k\neq
j\end{subarray}}^{N}\omega_{k}q_{k}\right)\omega_{j}\mathcal{M}_{N}\left(1,\dots,n,\dots,N\right)$

Using the fact that

$\frac{d}{d\omega_{j}}\left(\lim_{\Delta_{j}\to
1}\omega_{j}^{\Delta_{j}-1}\right)=\frac{d}{d\omega_{j}}\theta(\omega_{j})=\delta(\omega_{j})$
(29)

where $\theta(\omega)$ is the Heaviside step function, we see that the
integral on $\omega_{j}$ on the right hand side gives us

$\displaystyle\left\langle\prod_{n=1}^{N}\lim_{\Delta_{j}\rightarrow
1}(\Delta_{j}-1)\mathcal{O}_{\Delta_{n}}\left(z_{n},\bar{z}_{n},\eta^{n}\right)\right\rangle$
$\displaystyle=\left(\prod_{\begin{subarray}{c}n=1\\\ n\neq
j\end{subarray}}^{N}\int
d\omega_{n}\;\omega_{n}^{\Delta_{n}-1}\right)\delta^{(4)}\left(\sum_{\begin{subarray}{c}k=1\\\
k\neq j\end{subarray}}^{N}\omega_{k}q_{k}\right)$ (30) $\displaystyle\hskip
56.9055pt\times\lim_{\omega_{j}\to
0}\omega_{j}\mathcal{M}_{N}\left(1,\dots,n,\dots,N\right)$

Using the soft limit (27) we get

$\begin{split}&\left\langle\prod_{n=1}^{N}\lim_{\Delta_{j}\rightarrow
1}(\Delta_{j}-1)\mathcal{O}_{\Delta_{n}}\left(z_{n},\bar{z}_{n},\eta^{n}\right)\right\rangle=\sum_{\begin{subarray}{c}i=1\\\
i\neq
j\end{subarray}}^{N}\omega_{i}\Bigg{\\{}\frac{z_{j-1,i}^{2}\bar{z}_{ji}}{z_{j-1,j}^{2}z_{ji}}+\frac{\bar{z}_{j-1,i}^{2}z_{ji}}{\bar{z}_{j-1,j}^{2}\bar{z}_{ji}}\delta^{4}\left(\eta^{j}\right)\Bigg{\\}}\\\
&\hskip 142.26378pt\times\left(\prod_{\begin{subarray}{c}n=1\\\ n\neq
j\end{subarray}}^{N}\int_{0}^{\infty}d\omega_{k}\omega_{k}^{\Delta_{k}-1}\right)\delta^{(4)}\left(\sum_{\begin{subarray}{c}k=1\\\
k\neq
j\end{subarray}}^{N}\omega_{k}q_{k}\right)\mathcal{M}_{N-1}\left(1,\dots,i,\dots,N\right)\\\
&=\sum_{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}^{n}\Bigg{\\{}\frac{z_{j-1,i}^{2}\bar{z}_{ji}}{z_{j-1,j}^{2}z_{ji}}+\frac{\bar{z}_{j-1,i}^{2}z_{ji}}{\bar{z}_{j-1,j}^{2}\bar{z}_{ji}}\delta^{8}\left(\eta^{j}\right)\Bigg{\\}}\Bigg{[}\prod_{\begin{subarray}{c}n=1\\\
n\neq
i\end{subarray}}^{N}\int_{0}^{\infty}d\omega_{k}\;\omega_{k}^{\Delta_{k}-1}\int_{0}^{\infty}d\omega_{i}\;\omega_{i}^{\Delta_{i}}\\\
&\hskip 256.0748pt\delta^{(4)}\left(\sum_{\begin{subarray}{c}k=1\\\ k\neq
j\end{subarray}}^{N}\omega_{k}q_{k}\right)\mathcal{M}_{N-1}\left(1,\dots,i,\dots,N\right)\Bigg{]}\\\
&=\sum_{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}^{n}\Bigg{\\{}\frac{z_{j-1,i}^{2}\bar{z}_{ji}}{z_{j-1,j}^{2}z_{ji}}+\frac{\bar{z}_{j-1,i}^{2}z_{ji}}{\bar{z}_{j-1,j}^{2}\bar{z}_{ji}}\delta^{8}\left(\eta^{j}\right)\Bigg{\\}}\left\langle\mathcal{O}_{\Delta_{1}}\left(z_{1},\bar{z}_{1},\eta_{1}\right),\cdots\mathcal{O}_{\Delta_{i}+1}\left(z_{i},\bar{z}_{i},\eta^{i}\right),\cdots\right\rangle\end{split}$

The Super-Ward identity that we get from the conformally supersoft theorem is

$\begin{split}&\left\langle
J_{1}(z,\bar{z},\eta)\mathcal{O}_{\Delta_{1}}(z_{1},\bar{z}_{1},\eta^{1})\cdots\mathcal{O}_{\Delta_{N}}\left(z_{N},\bar{z}_{N},\eta^{N}\right)\right\rangle\\\
&=\sum_{i=1}^{N}\left\\{\frac{\left(\bar{z}-\bar{z}_{i}\right)}{\left(z-z_{i}\right)}\frac{\left(z_{N}-z_{i}\right)^{2}}{\left(z_{N}-z\right)^{2}}+\frac{\left(z-z_{i}\right)}{\left(\bar{z}-z_{i}\right)}\frac{\left(\bar{z}_{N}-\bar{z}_{i}\right)^{2}}{\left(\bar{z}_{N}-\bar{z}\right)^{2}}\delta^{8}(\eta)\right\\}\\\
&\hskip
64.58313pt\times\left\langle\mathcal{O}_{\Delta_{1}}\left(z_{1},\bar{z}_{1},\eta^{1}\right),\cdots\mathcal{O}_{\Delta_{i}+1}\left(z_{i},\bar{z}_{i},\eta^{i}\right),\cdots,\mathcal{O}_{\Delta_{N}}\left(z_{N},\bar{z}_{N},\eta^{N}\right)\right\rangle\end{split}$
(31)

where

$J_{1}(z,\bar{z},\eta)=\lim_{\Delta\to
1}(\Delta-1)\mathcal{O}_{\Delta}(z,\bar{z},\eta)$

is the $\Delta\to 1$ soft operator. In the above soft factor we chose the
reference vector for polarisation of the soft particle to be the momentum
vector of $n$th particle. We leave this reference vector arbitrary which
corresponds to a point $\xi\in\mathcal{CS}^{2}$. The super-Ward identity then
takes the form

$\begin{split}&\left\langle
J_{1}(z,\bar{z},\eta)\mathcal{O}_{\Delta_{1}}(z_{1},\bar{z}_{1},\eta^{1})\cdots\mathcal{O}_{\Delta_{N}}\left(z_{N},\bar{z}_{N},\eta^{N}\right)\right\rangle\\\
&=\sum_{i=1}^{N}\left\\{\frac{\left(\bar{z}-\bar{z}_{i}\right)}{\left(z-z_{i}\right)}\frac{\left(\xi-
z_{i}\right)^{2}}{\left(\xi-z\right)^{2}}+\frac{\left(z-z_{i}\right)}{\left(\bar{z}-\bar{z}_{i}\right)}\frac{\left(\bar{\xi}-\bar{z}_{i}\right)^{2}}{\left(\bar{\xi}-\bar{z}\right)^{2}}\delta^{8}(\eta)\right\\}\\\
&\hskip
64.58313pt\times\left\langle\mathcal{O}_{\Delta_{1}}\left(z_{1},\bar{z}_{1},\eta^{1}\right),\cdots\mathcal{O}_{\Delta_{i}+1}\left(z_{i},\bar{z}_{i},\eta^{i}\right),\cdots,\mathcal{O}_{\Delta_{N}}\left(z_{N},\bar{z}_{N},\eta^{N}\right)\right\rangle\end{split}$
(32)

When we expand both sides in the grassmann variables $\eta^{i}$ and compare
coefficients, we get the Ward identity for soft graviton operator:

$\begin{split}\left\langle
J_{1}(z,\bar{z})\prod_{n=1}^{N}\mathcal{O}_{\Delta_{n},\ell_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle=\sum_{i=1}^{N}\frac{\left(\bar{z}-\bar{z}_{i}\right)}{\left(z-z_{i}\right)}&\frac{\left(\xi-
z_{i}\right)^{2}}{\left(\xi-z\right)^{2}}\langle\mathcal{O}_{\Delta_{1},\ell_{1}}\left(z_{1},\bar{z}_{1}\right),\\\
&\cdots\mathcal{O}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots,\mathcal{O}_{\Delta_{N},\ell_{N}}\left(z_{N},\bar{z}_{N}\right)\rangle\end{split}$
(33)

and

$\begin{split}\left\langle\bar{J}_{1}(z,\bar{z})\prod_{n=1}^{N}\mathcal{O}_{\Delta_{n},\ell_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle=\sum_{i=1}^{N}\frac{\left(z-z_{i}\right)}{\left(\bar{z}-\bar{z}_{i}\right)}&\frac{\left(\bar{\xi}-\bar{z}_{i}\right)^{2}}{\left(\bar{\xi}-\bar{z}\right)^{2}}\langle\mathcal{O}_{\Delta_{1},\ell_{1}}\left(z_{1},\bar{z}_{1}\right),\\\
&\cdots\mathcal{O}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots,\mathcal{O}_{\Delta_{N},\ell_{N}}\left(z_{N},\bar{z}_{N}\right)\rangle\end{split}$
(34)

where

$J_{1}(z,\bar{z})=\lim_{\Delta\to
1}(\Delta-1)\mathcal{O}_{\Delta,+2}(z,\bar{z}),\quad\bar{J}_{1}(z,\bar{z})=\lim_{\Delta\to
1}(\Delta-1)\mathcal{O}_{\Delta,-2}(z,\bar{z})$ (35)

are the $\Delta=1$ soft graviton operators. The subleading soft factor was
also calculated in tab . It turns out that it is same as the subleading soft
factor for positive and negative helicity graviton in pure gravity
(He:2014bga, , Eq. (2.9)). We then write the super Ward identity following the
calculations in Adamo:2019ipt :

$\begin{split}&\left\langle
J_{0}(z,\bar{z},\eta)\mathcal{O}_{\Delta_{1}}(z,\bar{z},\eta^{1})\cdots\mathcal{O}_{\Delta_{N}}\left(z_{N},\bar{z}_{N},\eta^{N}\right)\right\rangle\\\
&=\sum_{i=1}^{N}\left\\{\frac{\left(\bar{z}-\bar{z}_{i}\right)}{\left(z-z_{i}\right)}\frac{\left(\xi-
z_{i}\right)}{\left(\xi-z\right)}((\bar{z}-\bar{z}_{i})\partial_{\bar{z}_{i}}-2\bar{h}_{i})+\frac{\left(z-z_{i}\right)}{\left(\bar{z}-\bar{z}_{i}\right)}\frac{\left(\bar{\xi}-\bar{z}_{i}\right)}{\left(\bar{\xi}-\bar{z}\right)}\delta^{8}(\eta)((z-z_{i})\partial_{z_{i}}-2h_{i})\right\\}\\\
&\hskip
85.35826pt\times\left\langle\mathcal{O}_{\Delta_{1}}\left(z_{1},\bar{z}_{1},\eta^{1}\right),\cdots\mathcal{O}_{\Delta_{i}}\left(z_{i},\bar{z}_{i},\eta^{i}\right)\cdots,\mathcal{O}_{\Delta_{N}}\left(z_{N},\bar{z}_{N},\eta^{N}\right)\right\rangle\end{split}$
(36)

where

$J_{0}(z,\bar{z},\eta)=\lim_{\Delta\to
0}\Delta[\mathcal{O}_{\Delta,+2}(z,\bar{z})+\delta^{8}(\eta)\mathcal{O}_{\Delta,-2}(z,\bar{z})].$

only contains the $\Delta=0$ soft graviton operators. This immediately gives
us the subleading soft graviton limit:

$\begin{split}\left\langle
J_{0}(z,\bar{z})\prod_{n=1}^{N}\mathcal{O}_{\Delta_{n},\ell_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle=\sum_{i=1}^{N}\frac{\left(\bar{z}-\bar{z}_{i}\right)}{\left(z-z_{i}\right)}\frac{\left(\xi-
z_{i}\right)}{\left(\xi-z\right)}&((\bar{z}-\bar{z}_{i})\partial_{\bar{z}_{i}}-2\bar{h}_{i})\\\
&\times\left\langle\cdots\mathcal{O}_{\Delta_{i},\ell_{i}}\left(z_{i},\bar{z}_{i}\right)\cdots\right\rangle\end{split}$
(37)

and

$\begin{split}\left\langle\bar{J}_{0}(z,\bar{z})\prod_{n=1}^{N}\mathcal{O}_{\Delta_{n},\ell_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle=\sum_{i=1}^{N}\frac{\left(z-z_{i}\right)}{\left(\bar{z}-\bar{z}_{i}\right)}\frac{\left(\bar{\xi}-\bar{z}_{i}\right)}{\left(\bar{\xi}-\bar{z}\right)}&((z-z_{i})\partial_{z_{i}}-2h_{i})\\\
&\times\left\langle\cdots\mathcal{O}_{\Delta_{i},\ell_{i}}\left(z_{i},\bar{z}_{i}\right)\cdots\right\rangle\end{split}$
(38)

where

$J_{0}(z,\bar{z})=\lim_{\Delta\to
0}\Delta\mathcal{O}_{\Delta,+2}(z,\bar{z}),\quad\bar{J}_{0}(z,\bar{z})=\lim_{\Delta\to
0}\Delta\mathcal{O}_{\Delta,-2}(z,\bar{z})$ (39)

are the $\Delta=0$ soft graviton operators and
$h_{i}=\frac{\Delta_{i}+\ell_{i}}{2},\bar{h}_{i}=\frac{\Delta_{i}-\ell_{i}}{2}$
are the conformal weights of the operator
$\mathcal{O}_{\Delta_{i},\ell_{i}}(z,\bar{z})$.

####  2.3.2 Soft gravitino operator

Next we move to soft gravitino operator. The leading soft gravitino limit for
superamplitudes is given by

$\mathcal{M}_{N+1}(\psi_{s+}^{A},\\{p_{1},\eta^{1}\\},\dots\\{p_{N},\eta^{N}\\})=\sum_{i=1}^{N}\frac{[si]\langle
ri\rangle}{\langle si\rangle\langle
rs\rangle}\frac{\partial}{\partial\eta_{iA}}\mathcal{M}_{N}(\\{p_{1},\eta^{1}\\},\dots\\{p_{N},\eta^{N}\\}),$
(40)

where $r$ is the reference vector corresponding to point
$\xi\in\mathcal{CS}^{2}$. The negative helicity soft gravitino limit can be
obtained by conjugating the soft factor. We can expand both sides in
$\eta^{i}$ and get the soft theorem in terms of component fields. Note that
because of $\partial/\partial_{\eta_{iA}}$, the soft gravitino operator
changes the spin of the particle
$\ell_{i}\to\ell_{i}^{c}\equiv\ell_{i}-\frac{1}{2}$. Thus we can only have

$\ell_{i}\in\\{-3/2,-1,-1/2,0,+1/2,+1,+3/2,+2\\}.$ (41)

For negative helicity gravitino $\ell_{i}^{c}\to\ell_{i}$ and clearly

$\ell_{i}^{c}\in\\{-2,-3/2,-1,-1/2,0,+1/2,+1,+3/2\\}.$ (42)

The explicit soft theorem in terms of celestial amplitudes is given333we have
put the R-symmetry index $*_{i}$ as a superscript for brevity but it can also
be on subscript depending on the helicity of the operator. Here the number of
fermions preceding particle $i$, $\sigma_{i}=1$ if
$\ell_{i}\in\mathbb{Z}+\frac{1}{2}$ and $0$ otherwise (see Fotopoulos:2020bqj
for detailed explanation). by (c.f. (Fotopoulos:2020bqj, , (6.2), (6.3)))

$\begin{split}\left\langle
J_{1/2}^{A}(z,\bar{z})\prod_{n=1}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle=\sum_{i=1}^{N}f(A,\ell_{i},*_{i},*^{\prime}_{i})&(-1)^{\sigma_{i}}\frac{\left(\bar{z}-\bar{z}_{i}\right)}{\left(z-z_{i}\right)}\frac{\left(\xi-
z_{i}\right)}{\left(\xi-z\right)}\\\
&\langle\cdots\mathcal{O}^{*^{\prime}_{i}}_{\Delta_{i}+\frac{1}{2},\ell_{i}^{c}}\left(z_{i},\bar{z}_{i}\right),\cdots\rangle\end{split}$
(43)

and

$\begin{split}\left\langle\bar{J}_{1/2\leavevmode\nobreak\
A}(z,\bar{z})\prod_{n=1}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}^{c}}\left(z_{n},\bar{z}_{n}\right)\right\rangle=\sum_{i=1}^{N}\bar{f}(A,\ell_{i}^{c},*_{i},*^{\prime}_{i})&(-1)^{\sigma_{i}}\frac{\left(z-z_{i}\right)}{\left(\bar{z}-\bar{z}_{i}\right)}\frac{\left(\bar{\xi}-\bar{z}_{i}\right)}{\left(\bar{\xi}-\bar{z}\right)}\\\
&\langle\cdots\mathcal{O}^{*^{\prime}_{i}}_{\Delta_{i}+\frac{1}{2},\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots\rangle\end{split}$
(44)

where

$J^{A}_{1/2}(z,\bar{z})=\lim_{\Delta\to\frac{1}{2}}\left(\Delta-\frac{1}{2}\right)\mathcal{O}^{A}_{\Delta,+\frac{3}{2}}(z,\bar{z}),\quad\bar{J}_{1/2\leavevmode\nobreak\
A}(z,\bar{z})=\lim_{\Delta\to\frac{1}{2}}\left(\Delta-\frac{1}{2}\right)\mathcal{O}_{\Delta,-\frac{3}{2},A}(z,\bar{z})$
(45)

are soft gravitino operators. The factors
$f(A,\ell_{i},*_{i},*^{\prime}_{i}),\bar{f}(A,\ell_{i}^{c},*_{i},*^{\prime}_{i})$
are the R-symmetry factors that we can determine using the collinear limits
given above. From (43) and (44) it is clear that the first argument of $f$ is
the R-symmetry index of the soft gravitino operator itself, the second and
third arguments are the helicity $\ell_{i}$ and R-symmetry index $*_{i}$
respectively of the operator $\mathcal{O}^{*_{i}}_{\Delta_{i},\ell_{i}}$ which
the soft gravitino will act on. Lastly the fourth argument will be the
R-symmetry index $*^{{}^{\prime}}_{i}$ of resultant operator. Similarly it
goes for $\bar{f}$. As an example, we can see from the OPE in Eq.(112) that
when $\ell_{i}=-\frac{3}{2}$, $f(A,-3/2,B,*^{\prime}_{i})=\delta^{A}_{B}$.
Since the resulting particle $\ell=-2$ has no R-symmetry index the
$*^{\prime}_{i}$ entry is empty.

The soft graviphoton limit can be calculated using the OPEs of graviphoton
operator with various conformal operators. These OPEs are listed in Appendix
A. Soft limits correspond to the values of scaling dimension $\Delta$ of the
graviphoton operator for which the beta functions appearing in the OPEs has
poles. From Appendix A we see that the OPEs of graviphoton operator with
various other operators involve444The OPE of two graviphoton operators with
opposite helicity involves another term, c.f. Eq. (113). One of the terms in
the OPE vanishes depending on which of the two helicities of the graviphoton
we take to be soft. See Appendix B for such calculations. $B(\Delta,*)$. Since
$B(\Delta,*)$ has poles at all non-positive integer values of $\Delta$, the
leading soft limit of the graviphoton operator is $\Delta\to 0$ and all other
negative integral values are subleading. In Subsection 3.2, we will need the
leading soft graviphoton limit.

Finally, as noted in tab , graviphotino and scalars are trivial in the soft
limit and hence do not correspond to any global symmetry Fotopoulos:2020bqj .
So we do not consider them further.

##  3 Asymptotic Symmetry Generators in $\mathcal{N}=8$ SUGRA

Let us first consider the obvious global symmetries of $\mathcal{N}=8$
supergravity. The global symmetry algebra consists of the Poincaré algebra and
the $\mathcal{N}=8$ supersymmetry algebra, together called the $\mathcal{N}=8$
super-Poincaré algebra and $\mathrm{SU}(8)_{R}$ $R-$symmetry algebra. At null
infinity, we expect to obtain infinite dimensional extensions of these
algebras. Following previous works Fotopoulos:2019vac ; Fotopoulos:2020bqj ;
Banerjee:2021uxe , we can easily construct the currents that extend the super-
Poincaré algebra, we call this algebra the $\mathcal{N}=8\leavevmode\nobreak\
\mathfrak{sbms}_{4}$ algebra. We start by constructing the currents for the
$\mathcal{N}=8\leavevmode\nobreak\ \mathfrak{sbms}_{4}$ algebra.

###  3.1 $\mathcal{N}=8\leavevmode\nobreak\ \mathfrak{sbms}_{4}$ algebra
currents

The $\mathfrak{bms}_{4}$ part of the $\mathcal{N}=8\leavevmode\nobreak\
\mathfrak{sbms}_{4}$ algebra is known to be generated Fotopoulos:2019vac  by
the shadow transform of the $\Delta=0$ graviton operator suitably modified as
discussed below. This is called the generator of superrotations and the level
one descendant of the $\Delta=1$ graviton operator is called the generator of
supertranslations on the celestial sphere. Let us define the shadow transforms
$T_{0}(z,\bar{z})$ and $\overline{T}_{0}(z,\bar{z})$ as:

$\displaystyle T_{0}(z,\bar{z})=\lim_{\Delta\to 0}\frac{3!\Delta}{2\pi}\int
d^{2}z^{\prime}\frac{1}{\left(z-z^{\prime}\right)^{4}}\mathcal{O}_{\Delta,-2}\left(z^{\prime},\bar{z}^{\prime}\right)$
(46) $\displaystyle\overline{T}_{0}(z,\bar{z})=\lim_{\Delta\to
0}\frac{3!\Delta}{2\pi}\int
d^{2}z^{\prime}\frac{1}{\left(\bar{z}-\bar{z}^{\prime}\right)^{4}}\mathcal{O}_{\Delta,+2}\left(z^{\prime},\bar{z}^{\prime}\right)$

It has been argued in Banerjee:2022wht  that the above shadow transform
operator does not satisfy the usual OPE of a stress tensor. In particular the
$T_{0}T_{0}$ OPE has an extra term which does not vanish as shown in
(Banerjee:2022wht, , Appendix C) unless we modify the stress tensor
appropriately. The origin of the problem is the observation that $T_{0}(z)$ is
not holomorphic:

$\bar{\partial}T_{0}=-\frac{1}{2}\partial^{3}\bar{J}_{0}(z,\bar{z})$ (47)

where $J_{0}$ is the $\Delta=0$ soft graviton operator defined in (39). Hence
the modified the stress tensor can be defined as follows:

$T_{\text{mod}}:=T_{0}+\frac{1}{2}\partial^{3}\epsilon_{\bar{J}_{0}}$ (48)

where

$\epsilon_{\bar{J}_{0}}:=\int_{\bar{z}_{0}}^{\bar{z}}d\bar{w}\bar{J}_{0}(z,\bar{w})$
(49)

with $z_{0}$ as a reference point. Then it has been shown that the modified
stress tensor satisfies the correct $T_{\text{mod}}T_{\text{mod}}$ OPE
(Banerjee:2022wht, , Appendix C). From now on we omit the subscript “mod” and
$T,\overline{T}$ will denote the modified stress tensor. Using the soft limits
(33), (34), (37) and (38) and performing the same calculations as in
Fotopoulos:2019vac , we arrive at the OPE

$\begin{split}&T(z)\mathcal{O}_{\Delta,\ell}(w,\bar{w})=\frac{h}{(z-w)^{2}}\mathcal{O}_{\Delta,\ell}(w,\bar{w})+\frac{1}{z-w}\partial_{w}\mathcal{O}_{\Delta,\ell}(w,\bar{w})+\text{
regular.}\\\
&\overline{T}(\bar{z})\mathcal{O}_{\Delta,\ell}(w,\bar{w})=\frac{\bar{h}}{(\bar{z}-\bar{w})^{2}}\mathcal{O}_{\Delta,\ell}(w,\bar{w})+\frac{1}{\bar{z}-\bar{w}}\partial_{\bar{w}}\mathcal{O}_{\Delta,\ell}(w,\bar{w})+\operatorname{regular.}\end{split}$
(50)

The supertranslations generator $P(z)$, $\overline{P}(z)$ are defined as:

$\displaystyle P(z)=\lim_{\Delta\rightarrow
1}\frac{(\Delta-1)}{4}\partial_{\bar{z}}\mathcal{O}_{\Delta,+2}(z,\bar{z})$
(51) $\displaystyle\overline{P}(\bar{z})=\lim_{\Delta\rightarrow
1}\frac{(\Delta-1)}{4}\partial_{z}\mathcal{O}_{\Delta,-2}(z,\bar{z}).$

For $P(z)$ we have

$\displaystyle P(z)\mathcal{O}_{\Delta,\ell}(w,\bar{w})$
$\displaystyle=\frac{1}{z-w}\mathcal{O}_{\Delta+1,\ell}(w,\bar{w})+\text{
regular }$ (52)

and similar OPEs hold for $\overline{P}(\bar{z})$ with conjugated poles. These
operators are related to the supertranslation generator
$\mathcal{P}(z,\bar{z})$, which is a primary field operator of conformal
weight $(\frac{3}{2},\frac{3}{2})$. By contour integrals Fotopoulos:2019vac :

$P(z)=\frac{1}{2\pi i}\oint
d\bar{z}\mathcal{P}(z,\bar{z}),\quad\overline{P}(\bar{z})=\frac{1}{2\pi
i}\oint dz\mathcal{P}(z,\bar{z}).$ (53)

The supertranslation satisfies the OPE

$\mathcal{P}(z,\bar{z})\mathcal{O}_{\Delta,\ell}(w,\bar{w})=\frac{1}{z-w}\frac{1}{\bar{z}-\bar{w}}\mathcal{O}_{\Delta+1,\ell}(w,\bar{w})+\text{regular.}$
(54)

The supercurrent for $\mathcal{N}=1$ supersymmetry was constructed in
Fotopoulos:2020bqj . We will see that the same construction will give us the 8
supercurrents for $\mathcal{N}=8$ supersymmetry. We thus define the
supercurrents as the shadow transform of the $\Delta=\frac{1}{2}$ gravitino
operator:

$\begin{split}&S_{A}(z)=\lim_{\Delta\to\frac{1}{2}}\frac{\Delta-\frac{1}{2}}{\pi}\int
d^{2}z^{\prime}\frac{1}{\left(z-z^{\prime}\right)^{3}}\mathcal{O}_{A;\Delta,-\frac{3}{2}}(z^{\prime},\bar{z}^{\prime})\\\
&\overline{S}^{A}(\bar{z})=\lim_{\Delta\to\frac{1}{2}}\frac{\Delta-\frac{1}{2}}{\pi}\int
d^{2}z^{\prime}\frac{1}{\left(\bar{z}-\bar{z}^{\prime}\right)^{3}}\mathcal{O}^{A}_{\Delta,+\frac{3}{2}}(z^{\prime},\bar{z}^{\prime})\end{split}$
(55)

Note that the above operators are also not holomorphic since

$\begin{split}\bar{\partial}S_{A}(z,\bar{z})=\lim_{\Delta\to
1/2}(\Delta-\frac{1}{2})\partial_{z}^{2}\mathcal{O}_{A;\Delta,-\frac{3}{2}}(z,\bar{z})=\partial^{2}\bar{J}_{1/2\;A}(z,\bar{z})\neq
0,\end{split}$ (56)

where $\bar{J}_{1/2\;A}(z,\bar{z})$ is the leading soft gravitino operator
defined in (45). One can modify it in a similar way as in Eq. (48). Put

$\epsilon_{\bar{J}_{1/2}\;A}(z,\bar{z}):=\int_{\bar{z}_{0}}^{z}d\bar{w}\bar{J}_{1/2\;A}(w,\bar{w})$
(57)

where $z_{0}$ is a reference point and define

$S^{A}_{\text{mod}}:=S^{A}-\partial^{2}\epsilon^{A}_{\bar{J}_{1/2}}.$ (58)

We emphasize that this modification is not required at the quantum level since
the OPEs of $S^{A}$ are as expected for a supercurrent. So we continue to use
the shadow transform of the leading soft gravitino operator as the
supercurrent without any modification.

Following the calculations of (Fotopoulos:2020bqj, , Section 7), it is
straightforward to see that

$\begin{split}&T(z)S_{A}(w)=\frac{3}{2}\frac{S_{A}(w)}{(z-w)^{2}}+\frac{\partial
S_{A}(w)}{z-w}+\text{ regular, }\\\
&\overline{T}(\bar{z})\overline{S}^{A}(\bar{w})=\frac{3}{2}\frac{\overline{S}^{A}(\bar{z})}{(\bar{z}-\bar{w})^{2}}+\frac{\bar{\partial}\overline{S}^{A}(\bar{w})}{\bar{z}-\bar{w}}+\text{
regular. }\end{split}$ (59)

and the OPEs $T\overline{S}^{A}$ and $\overline{T}S_{A}$ are regular. These
OPEs confirm the conformal weights of $S_{A}$ and $\overline{S}_{A}$ as
$(\frac{3}{2},0)$ and $(0,\frac{3}{2})$ respectively. We now want to show that

$:\\{S_{B}(z),\overline{S}^{A}(\bar{z})\\}:\leavevmode\nobreak\
=\leavevmode\nobreak\
:S_{B}(z)\overline{S}^{A}(\bar{z})+\overline{S}^{A}(\bar{z})S_{B}(z):\leavevmode\nobreak\
=\delta^{A}_{B}\mathcal{P}(z,\bar{z}).$ (60)

Using the gravitino soft limit (43) and (44) and the leading graviton limits
(33) and (34) and following the calculations in (Fotopoulos:2020bqj, , Section
7.3), we get555note that we do not seperate the operators in the correlator
according to their spins $\ell,\ell^{c}$ unlike Fotopoulos:2020bqj  since
there is an overlap in the ranges of the two spins. So in the correlators in
this calculation, the spins are assumed to be arbitrary.

$\begin{split}&\left\langle
S_{B}(z)\overline{S}^{A}(\bar{w})\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=\delta^{A}_{B}\sum_{i=3}^{N}\left[\frac{1}{(\bar{w}-\bar{z})^{2}}\frac{\bar{z}-\bar{z}_{i}}{z-z_{i}}+\frac{1}{\bar{z}-\bar{w}}\frac{1}{z-z_{i}}+\frac{1}{\bar{w}-\bar{z}_{i}}\frac{1}{z-z_{i}}\right]\left\langle\cdots\mathcal{O}^{*_{i}}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots\right\rangle\\\
&-\sum_{i=3}^{N}f(A,\ell_{i},*_{i},*_{i}^{\prime})\bar{f}(B,\ell_{i}-1/2,*_{i}^{\prime},*_{i}^{\prime\prime})\frac{1}{z-z_{i}}\frac{1}{\bar{w}-\bar{z}_{i}}\left\langle\cdots\mathcal{O}^{*_{i}^{\prime\prime}}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots\right\rangle\\\
&+\sum_{\begin{subarray}{c}i,j=3\\\ i\neq
j\end{subarray}}^{N}(-1)^{\sigma_{i}+\sigma_{j}}f(A,\ell_{i},*_{i},*_{i}^{\prime})\bar{f}(B,\ell_{j},*_{j},*_{j}^{\prime})\frac{1}{z-z_{i}}\frac{1}{\bar{w}-\bar{z}_{j}}\left\langle\cdots\mathcal{O}^{*_{i}^{\prime}}_{\Delta_{i}+\frac{1}{2},\ell_{i}-\frac{1}{2}}\left(z_{i},\bar{z}_{i}\right),\cdots\right.,\\\
&\hskip
284.52756pt\left.\mathcal{O}^{*_{j}^{\prime}}_{\Delta_{j}+\frac{1}{2},\ell_{j}+\frac{1}{2}}\left(z_{j},\bar{z}_{j}\right)\cdots\right\rangle\end{split}$
(61)

where the factors
$f(A,\ell_{i},*_{i},*_{i}^{\prime}),\bar{f}(B,\ell_{j},*_{j},*_{j}^{\prime})$
are the R-symmetry factors that appear on taking the soft or collinear limit
depending on the spins and helicities of the soft and collinear particles. In
this notation, the first argument of $f$ is the R-symmetry index of the
positive helicity soft gravitino, second argument is the spin (and helicity)
of one of the hard666that is not soft particles, the third argument is the
R-symmetry index of that hard particle (left implicit for generality) and the
fourth argument is the resulting R-symmetry index of the hard particle after
the soft limit is taken (again left implicit for generality). The notation for
$\bar{f}$ is similar. It is understood that if the spins do not belong to the
required range specified in (41) and (42) then $f,\bar{f}=0$. Similarly

$\begin{split}&\left\langle\overline{S}^{A}(\bar{z})S_{B}(w)\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=\delta^{A}_{B}\sum_{i=3}^{N}\left[\frac{1}{(w-z)^{2}}\frac{z-z_{i}}{\bar{z}-\bar{z}_{i}}+\frac{1}{z-w}\frac{1}{\bar{z}-\bar{z}_{i}}+\frac{1}{\bar{z}-\bar{z}_{i}}\frac{1}{w-z_{i}}\right]\left\langle\cdots\mathcal{O}^{*_{i}}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots\right\rangle\\\
&-\sum_{i=3}^{N}\bar{f}(B,\ell_{i},*_{i},*_{i}^{\prime})f(A,\ell_{i}+1/2,*_{i}^{\prime},*_{i}^{\prime\prime})\frac{1}{w-z_{i}}\frac{1}{\bar{z}-\bar{z}_{i}}\left\langle\cdots\mathcal{O}^{*_{i}^{\prime\prime}}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots\right\rangle\\\
&-\sum_{\begin{subarray}{c}i,j=3\\\ i\neq
j\end{subarray}}^{N}(-1)^{\sigma_{i}+\sigma_{j}}\bar{f}(B,\ell_{i},*_{i},*_{i}^{\prime})f(A,\ell_{j},*_{j},*_{j}^{\prime})\frac{1}{w-z_{i}}\frac{1}{\bar{z}-\bar{z}_{j}}\left\langle\cdots\mathcal{O}^{*_{i}^{\prime}}_{\Delta_{i}+\frac{1}{2},\ell_{i}+\frac{1}{2}}\left(z_{i},\bar{z}_{i}\right),\cdots\right.,\\\
&\hskip
284.52756pt\left.\mathcal{O}^{*_{j}^{\prime}}_{\Delta_{j}+\frac{1}{2},\ell_{j}-\frac{1}{2}}\left(z_{j},\bar{z}_{j}\right)\cdots\right\rangle\end{split}$
(62)

Thus the anticommutator is

$\begin{split}&\left\langle\left(\overline{S}^{A}(\bar{z})S_{B}(w)+S_{B}(z)\overline{S}^{A}(\bar{w})\right)\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=\delta^{A}_{B}\sum_{i=3}^{N}\left[\frac{1}{(w-z)^{2}}\frac{z-z_{i}}{\bar{z}-\bar{z}_{i}}+\frac{1}{z-w}\frac{1}{\bar{z}-\bar{z}_{i}}+\frac{1}{\bar{z}-\bar{z}_{i}}\frac{1}{w-z_{i}}\right]\left\langle\cdots\mathcal{O}^{*_{i}}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots\right\rangle\\\
&+\delta^{A}_{B}\sum_{i=3}^{N}\left[\frac{1}{(\bar{w}-\bar{z})^{2}}\frac{\bar{z}-\bar{z}_{i}}{z-z_{i}}+\frac{1}{\bar{z}-\bar{w}}\frac{1}{z-z_{i}}+\frac{1}{\bar{w}-\bar{z}_{i}}\frac{1}{z-z_{i}}\right]\left\langle\cdots\mathcal{O}^{*_{i}}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots\right\rangle\\\
&-\sum_{i=3}^{N}f(A,\ell_{i},*_{i},*_{i}^{\prime})\bar{f}(B,\ell_{i}-1/2,*_{i}^{\prime},*_{i}^{\prime\prime})\frac{1}{z-z_{i}}\frac{1}{\bar{w}-\bar{z}_{i}}\left\langle\cdots\mathcal{O}^{*_{i}^{\prime\prime}}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots\right\rangle\\\
&-\sum_{i=3}^{N}\bar{f}(B,\ell_{i},*_{i},*_{i}^{\prime})f(A,\ell_{i}+1/2,*_{i}^{\prime},*_{i}^{\prime\prime})\frac{1}{w-z_{i}}\frac{1}{\bar{z}-\bar{z}_{i}}\left\langle\cdots\mathcal{O}^{*_{i}^{\prime\prime}}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots\right\rangle.\end{split}$
(63)

Here in the last terms in Eq.(61) and Eq.(62) we have relative signs hence
both terms cancel. One can notice that relative sign is due to the action of
$S$ and $\overline{S}$ on different clusters for $i<j$ and $i>j$ in both the
terms. Then we see that the normal ordered current
$:\\{S_{B}(z),\overline{S}^{A}(\bar{z})\\}:$ satisfies

$\begin{split}&\left\langle:\\{S_{B}(z),\overline{S}^{A}(\bar{z})\\}:\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=2\delta^{A}_{B}\sum_{i=3}^{N}\frac{1}{\bar{z}-\bar{z}_{i}}\frac{1}{z-z_{i}}\left\langle\cdots\mathcal{O}^{*_{i}}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots\right\rangle\\\
&-\sum_{i=3}^{N}f(A,\ell_{i},*_{i},*_{i}^{\prime})\bar{f}(B,\ell_{i}-1/2,*_{i}^{\prime},*_{i}^{\prime\prime})\frac{1}{z-z_{i}}\frac{1}{\bar{z}-\bar{z}_{i}}\left\langle\cdots\mathcal{O}^{*_{i}^{\prime\prime}}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots\right\rangle\\\
&-\sum_{i=3}^{N}\bar{f}(B,\ell_{i},*_{i},*_{i}^{\prime})f(A,\ell_{i}+1/2,*_{i}^{\prime},*_{i}^{\prime\prime})\frac{1}{z-z_{i}}\frac{1}{\bar{z}-\bar{z}_{i}}\left\langle\cdots\mathcal{O}^{*_{i}^{\prime\prime}}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots\right\rangle.\end{split}$
(64)

We now show that for any $\ell_{i}$, R-symmetry factors in the last two sums
reduce to $\delta_{B}^{A}$. Let us start with $\ell_{i}=+2$ in which case
$*_{i},*_{i}^{\prime\prime}$ is empty. Moreover in this case
$\bar{f}(B,+2,*_{i},*_{i}^{\prime})=0$ so that we only have one term to
analyse. From the OPEs in Appendix A, we see that $*_{i}^{\prime}=A$ and

$f(A,+2,{}_{-},*_{i}^{\prime})\bar{f}(B,+3/2,*_{i}^{\prime},{}_{-})\mathcal{O}_{\Delta_{i}+1,+2}\left(z_{i},\bar{z}_{i}\right)=\delta^{A}_{B}\mathcal{O}_{\Delta_{i}+1,+2}\left(z_{i},\bar{z}_{i}\right).$
(65)

The case $\ell_{i}=+\frac{3}{2}$ is more interesting. Suppose $*_{i}=C$ then
from the OPEs, we can easily see that $*_{i}^{\prime}=AC$ for the second term
and $*_{i}^{\prime}$ is empty for the last term. We then have

$f(A,+3/2,C,*_{i}^{\prime})\bar{f}(B,+1,*_{i}^{\prime},*_{i}^{\prime\prime})\mathcal{O}^{*_{i}^{\prime\prime}}_{\Delta_{i}+1,+3/2}\left(z_{i},\bar{z}_{i}\right)=2!\delta^{[A}_{B}\mathcal{O}^{C]}_{\Delta_{i}+1,+3/2}\left(z_{i},\bar{z}_{i}\right)$
(66)

and similarly

$\bar{f}(B,+3/2,C,{}_{-})f(A,+2,{}_{-},*_{i}^{\prime\prime})\mathcal{O}^{*_{i}^{\prime\prime}}_{\Delta_{i}+1,+3/2}\left(z_{i},\bar{z}_{i}\right)=\delta^{C}_{B}\mathcal{O}^{A}_{\Delta_{i}+1,+3/2}\left(z_{i},\bar{z}_{i}\right)$
(67)

We can clearly see that the sum of the last two terms is simply
$\delta_{B}^{A}\mathcal{O}^{C}_{\Delta_{i}+1,+3/2}\left(z_{i},\bar{z}_{i}\right)$.
The case $\ell_{i}=-\frac{3}{2}$ is similar. Let us now analyse the case
$\ell_{i}=+1$ in which case $*_{i}=CD$. We get

$\begin{split}f(A,+1,CD,*_{i}^{\prime})\bar{f}(B,+1/2,*_{i}^{\prime},*_{i}^{\prime\prime})&\mathcal{O}^{*_{i}^{\prime\prime}}_{\Delta_{i}+1,+2}\left(z_{i},\bar{z}_{i}\right)\\\
&=\bar{f}(B,+1/2,*_{i}^{\prime},ACD)\mathcal{O}^{ACD}_{\Delta_{i}+\frac{1}{2},+1/2}\left(z_{i},\bar{z}_{i}\right)\\\
&=3\delta^{[A}_{B}\mathcal{O}^{CD]}_{\Delta_{i}+1,+1}\left(z_{i},\bar{z}_{i}\right).\end{split}$
(68)

Similarly

$\bar{f}(B,+1,CD,*_{i}^{\prime})f(A,+3/2,*_{i}^{\prime},*_{i}^{\prime\prime})\mathcal{O}^{*_{i}^{\prime\prime}}_{\Delta_{i}+1,+3/2}\left(z_{i},\bar{z}_{i}\right)=-2!\delta^{[C}_{B}\mathcal{O}^{D]A}_{\Delta_{i}+1,+1}\left(z_{i},\bar{z}_{i}\right)$
(69)

which finally implies

$\begin{split}3\delta^{[A}_{B}\mathcal{O}^{CD]}-2!\delta^{[C}_{B}\mathcal{O}^{D]A}&=\frac{1}{2}\left[(\delta^{A}_{B}\mathcal{O}^{CD}-\delta^{A}_{B}\mathcal{O}^{DC})+(\delta^{C}_{B}\mathcal{O}^{DA}-\delta^{C}_{B}\mathcal{O}^{AD})+(\delta^{D}_{B}\mathcal{O}^{AC}-\delta^{D}_{B}\mathcal{O}^{CA})\right]\\\
&-\delta^{C}_{B}\mathcal{O}^{DA}+\delta^{D}_{B}\mathcal{O}^{CA}\\\
&=\delta^{A}_{B}\mathcal{O}^{CD}\end{split}$ (70)

The case $\ell_{i}=-1$ is similar. The same calculation as in $\ell_{i}=1$
recurs for the cases $\ell_{i}=1/2,0$.

These calculations simplify the OPE (64). We get

$\left\langle:\\{S_{B}(z),\overline{S}^{A}(\bar{z})\\}:\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle=\delta^{A}_{B}\sum_{i=3}^{N}\frac{1}{\bar{z}-\bar{z}_{i}}\frac{1}{z-z_{i}}\left\langle\cdots\mathcal{O}^{*_{i}}_{\Delta_{i}+1,\ell_{i}}\left(z_{i},\bar{z}_{i}\right),\cdots\right\rangle$
(71)

In particular,

$:\\{S_{B}(z),\overline{S}^{A}(\bar{z})\\}:\mathcal{O}_{\Delta,\ell}(w,\bar{w})=\frac{1}{z-w}\frac{1}{\bar{z}-\bar{w}}\mathcal{O}_{\Delta+1,\ell}(w,\bar{w})+\text{regular.}$
(72)

Comparing this OPE with (54) readily implies the desired result

$:\\{S_{B}(z),\overline{S}^{A}(\bar{z})\\}:\leavevmode\nobreak\
=\delta^{A}_{B}\mathcal{P}(z,\bar{z}).$ (73)

###  3.2 Possible R-symmetry current

Recall that R-symmetry acts on supercharges $Q^{A}_{\alpha}$ and
$\overline{Q}_{\dot{\alpha}A},\leavevmode\nobreak\ A=1,\dots,8$ by multiplying
a unitary matrix $U\in U(8)$. This means that the supercharges transform in
the fundamental representation of R-symmetry group. At the level of Lie
algebra, we can identify the R-symmetry group as simply
$\mathfrak{su}(8)\oplus\mathfrak{u}(1)$ since
$\mathrm{U}(8)\cong(\mathrm{SU}(8)\times\mathrm{U}(1))/\mathbb{Z}_{8}$. Thus
we can label the generators of R-symmetry to be $T^{A}_{B}$ and $R$, where
$T^{A}_{B}$ are generators of the fundamental representation of
$\mathrm{SU}(8)$ satisfying the $\mathfrak{su}(8)$ algebra:

$\left[T^{A}_{B},T^{C}_{D}\right]=\delta_{D}^{A}T^{C}_{B}-\delta_{B}^{C}T^{A}_{D},$
(74)

and $R$ is the generator of the scaling U(1). A suitable matrix representation
for the generators is Bianchi:2008pu ,

$\left(T_{B}^{A}\right)^{C}{}_{D}=\delta_{D}^{A}\delta_{B}^{C}-\frac{1}{8}\delta_{B}^{A}\delta_{D}^{C}$
(75)

$T^{A}_{B}$ acts on the supercharges as

$\left[T^{A}_{B},Q^{C}_{\alpha}\right]=\left(T^{A}_{B}\right)^{C}_{\leavevmode\nobreak\
D}Q^{D}_{\alpha},\quad\left[T^{A}_{B},\overline{Q}_{\dot{\alpha}C}\right]=-\left(T^{A}_{B}\right)^{D}_{\leavevmode\nobreak\
C}\overline{Q}_{\dot{\alpha}D}.$ (76)

We now want to construct a current
$\widetilde{\mathcal{G}}^{A}_{B}(z,\bar{z})$ whose modes will extend the
generators $T^{A}_{B}$. As will be shown in Section 4, the modes of the
supercurrents $S_{A},\overline{S}^{A}$ will extend the supercharges. Since the
OPE of currents directly translates to the commutator of their modes within
radial quantisation, our currents must satisfy the OPE:

$\begin{split}&\widetilde{\mathcal{G}}^{A}_{B}(z,\bar{z})S_{C}(w)\sim((z-w)\leavevmode\nobreak\
\text{singularity})\left(T^{A}_{B}\right)^{D}_{\leavevmode\nobreak\
C}S_{D}(w),\\\
&\widetilde{\mathcal{G}}^{A}_{B}(z,\bar{z})\overline{S}^{C}(w)\sim-((\bar{z}-\bar{w})\leavevmode\nobreak\
\text{singularity})\left(T^{A}_{B}\right)^{C}_{\leavevmode\nobreak\
D}\overline{S}^{D}(w).\end{split}$ (77)

Note that $S_{A},\overline{S}^{A}$ are holomorphic and antiholomorphic
currents respectively, this imposes the condition that the singularities in
(77) be holomorphic and antiholomorphic respectively. As will be shown in
Section 4, nonholomorphic (holomorphic) singularity in the OPE of
$\mathcal{G}^{A}_{B}(z,\bar{z})$ with $S_{C}(w)$ ($\overline{S}^{C}(w)$)
results in nonsensical algebra. This requirement will be crucial.

The only conformal operator we are left with is the graviphoton operator.
Moreover, the leading soft graviphoton operator corresponds to $\Delta=0$ as
can be inferred from the poles of the beta function in the OPEs of graviphoton
operators with other operators which are summarised in Appendix A. It is clear
that we must consider the order independent graviphoton double soft limit with
opposite helicity, that is their normal ordered commutator (since they are
bosonic). Since it contains the factor $\delta^{AB}_{CD}$, as can be seen from
the collinear limits, this can be manipulated properly to obtain the
$\mathrm{SU}(8)$ generators. Here we consider the most general integral
transform corresponding to negative and positive helicity soft graviphotons
respectively as,

$\begin{split}&G_{AB}(z,\bar{z})=\lim_{\Delta\to 0}\frac{\Delta}{\pi}\int
d^{2}z^{\prime}\frac{1}{\left(z-z^{\prime}\right)^{a}}\frac{1}{\left(\bar{z}-\bar{z}^{\prime}\right)^{b}}\mathcal{O}_{AB;\Delta,-1}(z^{\prime},\bar{z}^{\prime})\\\
&\overline{G}^{CD}(z,\bar{z})=\lim_{\Delta\to 0}\frac{\Delta}{\pi}\int
d^{2}z^{\prime}\frac{1}{\left(\bar{z}-\bar{z}^{\prime}\right)^{a^{\prime}}}\frac{1}{\left(z-z^{\prime}\right)^{b^{\prime}}}\mathcal{O}^{CD}_{\Delta,+1}(z^{\prime},\bar{z}^{\prime}).\end{split}$
(78)

One can easily see that we can recover the usual shadow transformation
Fan:2020xjj  by taking specific values of $a$ and $b$. The operators
$\mathcal{O}_{AB;0,-1}$ and $\mathcal{O}^{CD}_{0,+1}$ have conformal weights
$(-\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},-\frac{1}{2})$ respectively.
Hence scaling transformation reveals the conformal weights of the currents
$G_{AB}$ and $\overline{G}^{CD}$ to be $(a-\frac{3}{2},b-\frac{1}{2})$ and
$(b^{\prime}-\frac{1}{2},a^{\prime}-\frac{3}{2})$ respectively.
Let us start with the OPE of our new currents $G_{AB}$ and $\overline{G}^{CD}$
with any conformal primary operators.

$\begin{split}&\left\langle
G_{AB}(z,\bar{z})\prod_{n=2}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=\lim_{\begin{subarray}{c}\Delta_{1}\to
0\end{subarray}}\frac{\Delta_{1}}{\pi}\int
d^{2}z_{1}\frac{1}{\left(z-z_{1}\right)^{a}}\frac{1}{\left(\bar{z}-\bar{z}_{1}\right)^{b}}\left\langle\mathcal{O}_{AB\leavevmode\nobreak\
\Delta_{1},-1}(z_{1},\bar{z}_{1})\prod_{n=2}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=\frac{1}{\pi}\int
d^{2}z_{1}\frac{1}{\left(z-z_{1}\right)^{a}}\frac{1}{\left(\bar{z}-\bar{z}_{1}\right)^{b}}\Bigg{[}\sum_{n=2}^{N}f(A,B,\ell_{i},*_{n},*_{n}^{\prime})\frac{z_{1}-z_{n}}{\bar{z}_{1}-\bar{z}_{n}}\left\langle\cdots\mathcal{O}^{*^{\prime}_{n}}_{\Delta_{n},\ell_{n}+1}(z_{n},\bar{z}_{n})\right\rangle\Bigg{]}\end{split}$
(79)

where $f(A,B,\ell_{i},*_{n},*_{n}^{\prime})$ contains the R-symmetry index of
the operators in the correlation function which appears on taking the
collinear limit. We used the fact that $\lim_{\Delta\to
0}\Delta\;B(\Delta,*)=1$. Now we use two basic integrals (see (Jiang:2021ovh,
, Appendix B) for proof):

$\begin{split}&\int
d^{2}z_{1}\frac{1}{\left(z-z_{1}\right)^{A}}\frac{1}{\left(\bar{z}-\bar{z}_{1}\right)^{B}}\frac{\left(\bar{z}_{1}-\bar{z}_{j}\right)^{s}}{z_{1}-z_{j}}=C_{s}(A,B)\frac{1}{\left(z_{j}-z\right)^{A}\left(\bar{z}_{j}-\bar{z}\right)^{B-s-1}}\\\
&\int
d^{2}z_{1}\frac{1}{\left(\bar{z}-\bar{z}_{1}\right)^{A}}\frac{1}{\left(z_{1}-z_{1}\right)^{B}}\frac{\left(z_{1}-z_{j}\right)^{s}}{\bar{z}_{1}-\bar{z}_{j}}=C_{s}(A,B)\frac{1}{\left(\bar{z}_{j}-\bar{z}\right)^{A}}\frac{1}{\left(z_{j}-z\right)^{B-s-1}}\end{split}$
(80)

where

$C_{s}(A,B)=\frac{(-1)^{s+A+B}(-\pi)s!}{(-B+1)(-B+2)\cdots(-B+s+1)}$ (81)

Now performing the shadow integral for $n\neq 1$ and $s=1$,

$\begin{split}\int
d^{2}z_{1}\frac{1}{\left(z-z_{1}\right)^{a}}\frac{1}{\left(\bar{z}-\bar{z}_{1}\right)^{b}}\frac{z_{1}-z_{n}}{\bar{z}_{1}-\bar{z}_{n}}=C_{1}(b,a)\frac{1}{(\bar{z}_{n}-\bar{z})^{b}}\frac{1}{(z_{n}-z)^{a-2}}\end{split}$
(82)

We have,

$\begin{split}&\left\langle
G_{AB}(z,\bar{z})\prod_{n=2}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=\sum_{i=2}^{N}f(A,B,\ell_{i},*_{i},*_{i}^{\prime})C_{1}(b,a)\frac{1}{(\bar{z}_{i}-\bar{z})^{b}}\frac{1}{(z_{i}-z)^{a-2}}\left\langle\cdots\mathcal{O}^{*^{\prime}_{i}}_{\Delta_{i},\ell_{i}+1}(z_{i},\bar{z}_{i})\right\rangle\end{split}$
(83)

Here the helicities of the conformal operators inside the correlator are
restricted to
$\ell_{n}\in\\{-2,-\frac{3}{2},-1,-\frac{1}{2},0,+\frac{1}{2},+1\\}$. This can
be verified from the beta function singularities in the OPEs in Appendix A.
Similarly we can have the OPE for antiholomorphic current $\overline{G}^{CD}$
which act on the conformal primaries with helicities restricted in the range
$\ell^{\prime}_{n}\in\\{-1,-\frac{1}{2},0,+\frac{1}{2},+1,+\frac{3}{2},+2\\}$,

$\begin{split}&\left\langle\overline{G}^{CD}(z,\bar{z})\prod_{n=2}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell^{\prime}_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=\sum_{n=2}^{N}\bar{f}(C,D,\ell_{i}^{\prime},*_{n},*_{n}^{\prime})C(b^{\prime},a^{\prime})\frac{1}{(\bar{z}_{n}-\bar{z})^{b^{\prime}}}\frac{1}{(z_{n}-z)^{a^{\prime}-2}}\left\langle\cdots\mathcal{O}^{*^{\prime}_{n}}_{\Delta_{n},\ell^{\prime}_{n}-1}(z_{n},\bar{z}_{n})\right\rangle\end{split}$
(84)

Here we can pair $\ell^{\prime}$ with $\ell=\ell^{\prime}+1$. Hence we can
write the OPEs as,

$\begin{split}&G_{AB}(z)\mathcal{O}^{*}_{\Delta,\ell}(w,\bar{w})\sim
f(A,B,\ell,*,*^{\prime})C_{1}(b,a)\frac{1}{(\bar{w}-\bar{z})^{b}}\frac{1}{(w-z)^{a-2}}\mathcal{O}^{*^{\prime}}_{\Delta,\ell^{\prime}}(z,\bar{w})\\\
&\overline{G}^{CD}(z)\mathcal{O}^{*}_{\Delta,\ell^{\prime}}(w,\bar{w})\sim\bar{f}(C,D,\ell^{\prime},*,*^{\prime})C_{1}(b^{\prime},a^{\prime})\frac{1}{(\bar{w}-\bar{z})^{b^{\prime}}}\frac{1}{(z-z)^{a^{\prime}-2}}\mathcal{O}^{*^{\prime}}_{\Delta,\ell}(w,\bar{w})\end{split}$
(85)

####  3.2.1 The composite current

To construct a suitable current for R-symmetry, we need to use double soft
limits of the graviphoton operators. As is well known, double soft limit of
opposite helicity operators depend on the order of the soft limit. For this
reason, as in Banerjee:2021uxe  we consider the following operator:

$\mathcal{G}_{AB}^{CD}(z,\bar{z};w,\bar{w}):=G_{AB}(z)\overline{G}^{CD}(\bar{w})-\overline{G}^{CD}(\bar{w})G_{AB}(z)\equiv\Big{[}G_{AB}(z),\overline{G}^{CD}(\bar{w})\Big{]}.$
(86)

In order to construct a local operator, one needs to consider the normal order
of this operator evaluated at $z=w,\bar{z}=\bar{z}$. We thus define

$\begin{split}\mathcal{G}_{AB}^{CD}(z,\bar{z})=\;:\mathcal{G}_{AB}^{CD}(z,\bar{z};z,\bar{z}):\;&=\;:G_{AB}(z)\overline{G}^{CD}(\bar{z})-\overline{G}^{CD}(\bar{z})G^{AB}(z):\\\
&\equiv\;:\Big{[}G_{AB}(z),\overline{G}^{CD}(\bar{z})\Big{]}:.\end{split}$
(87)

We show in Appendix B that subject to the requirement of the R-symmetry
current explained above (77), the current $\mathcal{G}_{AB}^{CD}(z,\bar{z})$
satisfies the following OPE:

$\begin{split}&\left\langle\mathcal{G}_{AB}^{CD}(z,\bar{z})\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=(-1)^{a+b+a^{\prime}+b^{\prime}}C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})\Bigg{[}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})\bar{f}(C,D,\ell_{j}+1,*_{j}^{\prime},*_{j}^{\prime\prime})\\\
&\hskip
28.45274pt\times\frac{1}{\left(z-z_{j}\right)^{a+b^{\prime}-2}}\frac{1}{\left(\bar{z}-\bar{z}_{j}\right)^{a^{\prime}+b-2}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*^{{}^{\prime\prime}}_{j}}_{\Delta_{j},\ell_{j}}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&\hskip
56.9055pt-\bar{f}(C,D,\ell_{j},*_{j},*_{j}^{\prime})f(A,B,\ell_{j}-1,*_{j}^{\prime},*_{j}^{\prime\prime})\\\
&\hskip
28.45274pt\times\frac{1}{\left(z-z_{j}\right)^{a^{\prime}+b-2}}\frac{1}{\left(\bar{z}-\bar{z}_{j}\right)^{a+b^{\prime}-2}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*^{{}^{\prime\prime}}_{j}}_{\Delta_{j},\ell_{j}}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\Bigg{]}\end{split}$
(88)

In particular,

$\begin{split}&\mathcal{G}_{AB}^{CD}(z,\bar{z})\mathcal{O}_{E\leavevmode\nobreak\
\Delta,-\frac{3}{2}}(w,\bar{w})\sim-\delta^{CD}_{AB}\frac{(-1)^{a+b+a^{\prime}+b^{\prime}}C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})}{\left(z-w\right)^{a+b^{\prime}-2}\left(\bar{z}-\bar{w}\right)^{a^{\prime}+b-2}}\mathcal{O}_{E\leavevmode\nobreak\
\Delta,-\frac{3}{2}}(w,\bar{w})\\\
&\mathcal{G}_{AB}^{CD}(z,\bar{z})\mathcal{O}^{E}_{\Delta,+\frac{3}{2}}(w,\bar{w})\sim\delta^{CD}_{AB}\frac{(-1)^{a+b+a^{\prime}+b^{\prime}}C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})}{\left(z-w\right)^{a^{\prime}+b-2}\left(\bar{z}-\bar{w}\right)^{a+b^{\prime}-2}}\mathcal{O}^{E}_{\Delta,+\frac{3}{2}}(w,\bar{w})\end{split}$
(89)

where we used the fact that for the gravitino operator, the R-symmetry factor
in the double soft limit is $-\delta^{CD}_{AB}$. Indeed

$\lim_{\Delta_{1}\to
0}\Delta_{1}\mathcal{O}_{AB;\Delta_{1},-1}(z,\bar{z})\mathcal{O}_{E,\Delta,-\frac{3}{2}}\left(z_{1},\bar{z}_{1}\right)=\frac{z-z_{1}}{\bar{z}-\bar{z}_{1}}\mathcal{O}_{ABE;\Delta,-\frac{1}{2}}\left(z_{1},\bar{z}_{1}\right)$

If $E=r$, $A=a$, $B=b$, $C=c$, $D=d$, then

$\lim_{\Delta_{2}\to
0}\Delta_{2}\mathcal{O}_{\Delta_{2,+1}}^{cd}(w,\bar{w})\mathcal{O}_{abr;\Delta,-\frac{1}{2}}\left(z_{1},\bar{z}_{1}\right)=-\delta_{ab}^{cd}\frac{\bar{w}-\bar{z}_{1}}{w-z_{1}}\mathcal{O}_{r,\Delta,-\frac{3}{2}}\left(z_{1},\bar{z}_{1}\right)$

In all other cases, one can check from the collinear limit in Appendix A that
the R-symmetry factor is $-\delta^{CD}_{AB}$. Let us construct a new current
as a linear combination of our previous currents as follows:

$\left(\widetilde{\mathcal{G}}^{A}_{B}\right)^{C}_{D}(z,\bar{z}):=-\Bigg{(}\frac{1}{7}\delta^{A}_{D}\sum_{E=1}^{8}\mathcal{G}^{EC}_{BE}(z,\bar{z})+\frac{1}{56}\delta_{B}^{A}\sum_{E=1}^{8}\mathcal{G}^{EC}_{ED}(z,\bar{z})\Bigg{)}.$
(90)

Using the definition of generalised Kronecker delta

$\delta^{a_{1}\dots a_{n}}_{b_{1}\dots b_{n}}=\sum_{\sigma\in
S_{n}}\text{sign}(\sigma)\delta^{a_{1}}_{b_{\sigma(1)}}\dots\delta^{a_{n}}_{b_{\sigma(n)}},$
(91)

we see that

$\sum_{E=1}^{8}\delta^{EC}_{BE}=-7\delta^{C}_{B},\quad\sum_{E=1}^{8}\delta^{EC}_{ED}=7\delta^{C}_{D}.$
(92)

This gives us the OPE

$\begin{split}&\left(\widetilde{\mathcal{G}}^{C}_{A}\right)^{D}_{B}(z,\bar{z})\mathcal{O}_{D\leavevmode\nobreak\
\Delta,-\frac{3}{2}}(w,\bar{w})\sim\frac{(-1)^{a+b+a^{\prime}+b^{\prime}}C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})}{\left(z-w\right)^{a+b^{\prime}-2}\left(\bar{z}-\bar{w}\right)^{a^{\prime}+b-2}}\left(T_{A}^{C}\right)_{B}^{D}\mathcal{O}_{D\leavevmode\nobreak\
\Delta,-\frac{3}{2}}(w,\bar{w})\\\
&\left(\widetilde{\mathcal{G}}^{C}_{A}\right)^{D}_{B}(z,\bar{z})\mathcal{O}^{B}_{\Delta,+\frac{3}{2}}(w,\bar{w})\sim-\frac{(-1)^{a+b+a^{\prime}+b^{\prime}}C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})}{\left(z-w\right)^{a^{\prime}+b-2}\left(\bar{z}-\bar{w}\right)^{a+b^{\prime}-2}}\left(T_{A}^{C}\right)_{B}^{D}\mathcal{O}^{B}_{\Delta,+\frac{3}{2}}(w,\bar{w})\end{split}$
(93)

Hence $\widetilde{\mathcal{G}}^{C}_{A}$ is a candidate which can extend the
R-symmetry algebra. But we see an immediate problem. The OPE of
$\widetilde{\mathcal{G}}^{C}_{A}$ with supercurrents
$S_{D}(w),\overline{S}^{B}(\bar{w})$ is given by

$\begin{split}\left(\widetilde{\mathcal{G}}^{C}_{A}\right)^{D}_{B}(z,\bar{z})S_{D}(w)\sim&\left(T_{A}^{C}\right)_{B}^{D}\lim_{\Delta\rightarrow\frac{1}{2}}\frac{\Delta-\frac{1}{2}}{\pi}(-1)^{a+b+a^{\prime}+b^{\prime}}C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})\\\
&\times\int
d^{2}z_{1}\frac{1}{\left(w-z_{1}\right)^{3}}\frac{1}{\left(z-z_{1}\right)^{a+b^{\prime}-2}}\frac{1}{\left(\bar{z}-\bar{z}_{1}\right)^{a^{\prime}+b-2}}\mathcal{O}_{D,\Delta,-\frac{3}{2}}\left(z_{1},\bar{z}_{1}\right)\end{split}$
(94)

and

$\begin{split}\left(\widetilde{\mathcal{G}}^{C}_{A}\right)^{D}_{B}(z,\bar{z})\overline{S}^{B}(\bar{w})\sim&-\left(T_{A}^{C}\right)_{B}^{D}\lim_{\Delta\rightarrow\frac{1}{2}}\frac{\Delta-\frac{1}{2}}{\pi}(-1)^{a+b+a^{\prime}+b^{\prime}}C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})\\\
&\times\int
d^{2}z_{1}\frac{1}{\left(\bar{w}-\bar{z}_{1}\right)^{3}}\frac{1}{\left(z-z_{1}\right)^{a^{\prime}+b-2}}\frac{1}{\left(\bar{z}-\bar{z}_{1}\right)^{a+b^{\prime}-2}}\mathcal{O}^{B}_{\Delta,+\frac{3}{2}}\left(z_{1},\bar{z}_{1}\right)\end{split}$
(95)

The requirement (77) forces $a^{\prime}+b-2=0$ in (94) and (95). But then in
view of (136) we get

$a+b^{\prime}-2=0\quad\text{and}\quad a^{\prime}+b-2=0$ (96)

and conclude that the OPE is trivial

$\left(\widetilde{\mathcal{G}}^{C}_{A}\right)^{D}_{B}(z,\bar{z})S_{D}(w)\sim\text{regular},\quad\left(\widetilde{\mathcal{G}}^{C}_{A}\right)^{D}_{B}(z,\bar{z})\overline{S}^{B}(\bar{w})\sim\text{regular.}$
(97)

##  4 The $\mathcal{N}=8$ $\mathfrak{sbms}_{4}$ algebra

Let us now find out the asymptotic symmetries of the theory. The usual
symmetry currents in the theory are the stress tensor
$T(z),\overline{T}(\bar{z})$, which are the superrotation generators and
$\mathcal{P}(z,\bar{z})$, which is the supertranslation generator. The modes
of these currents generate the $\mathfrak{bms}_{4}$ algebra as described in
Fotopoulos:2019vac . As usual the generators of $\mathfrak{bms}_{4}$ are the
modes of $T(z),\overline{T}(\bar{z})$ and $\mathcal{P}(z,\bar{z})$. Let us
expand these currents in modes:

$T(z)=\sum_{n\in\mathbb{Z}}L_{n}z^{-n-2},\quad\overline{T}(\bar{z})=\sum_{n\in\mathbb{Z}}\bar{L}_{n}\bar{z}^{-n-2};$
(98)
$\begin{split}\mathcal{P}(z,\bar{z})\equiv\sum_{n,m\in\mathbb{Z}}P_{n-\frac{1}{2},m-\frac{1}{2}}z^{-n-1}\bar{z}^{-m-1}\end{split}$
(99)

As discussed in Fotopoulos:2019vac , the modes
$P_{n-\frac{1}{2},m-\frac{1}{2}}$ can be obtained from the modes of the
current $P(z)$ or $\overline{P}(\bar{z})$. If we write

$P(z)=\sum_{n\in\mathbb{Z}}P_{n-\frac{1}{2}}z^{-n-1},\quad\overline{P}(\bar{z})=\sum_{m\in\mathbb{Z}}\overline{P}_{m-\frac{1}{2}}z^{-m-1}$
(100)

then

$P_{n-\frac{1}{2},-\frac{1}{2}}=P_{n-\frac{1}{2}},\quad
P_{-\frac{1}{2},m-\frac{1}{2}}=\overline{P}_{m-\frac{1}{2}}$ (101)

and

$\begin{split}P_{n-\frac{1}{2},m-\frac{1}{2}}&=\frac{1}{i\pi(m+1)}\oint
d\bar{w}\bar{w}^{m+1}\left[\bar{T}(\bar{w}),P_{n-\frac{1}{2},-\frac{1}{2}}\right]\\\
&=\frac{1}{i\pi(m+1)}\oint
dww^{n+1}\left[T(w),\overline{P}_{-\frac{1}{2},m-\frac{1}{2}}\right].\end{split}$
(102)

These modes satisfy the usual $\mathfrak{bms}_{4}$ algebra:

$\begin{split}&\left[L_{m},L_{n}\right]=(m-n)L_{m+n},\quad\left[\bar{L}_{m},\bar{L}_{n}\right]=(m-n)\bar{L}_{m+n}\\\
&\left[L_{n},P_{kl}\right]=\left(\frac{1}{2}n-k\right)P_{n+k,l},\quad\left[\bar{L}_{n},P_{kl}\right]=\left(\frac{1}{2}n-l\right)P_{k,n+l},\end{split}$
(103)

where $m,n\in\mathbb{Z}$ and $k,l\in\mathbb{Z}+\frac{1}{2}$. In addition, an
infinite dimensional extension of the $\mathcal{N}=1$ supersymmetry algebra
was constructed in Fotopoulos:2020bqj . The supercurrent was shown to be the
shadow transform of the gravitino operator. In our theory, we have 8
supercurrents $S^{A}(z)$ and their antiholomorphic counterpart
$\overline{S}_{A}(\bar{z})$. The OPEs (59) show that $S^{A}(z)$ and
$\overline{S}_{A}(\bar{z})$ are conformal primaries of dimension
$(\frac{3}{2},0)$ and $(0,\frac{3}{2})$ respectively. Consequently if we
expand the supercurrents as

$\begin{split}&S_{A}(z)=\sum_{k\in\mathbb{Z}+\frac{1}{2}}\frac{(S_{A})_{k}}{z^{k+\frac{3}{2}}},\quad\text{with}\quad(S_{A})_{k}=\frac{1}{2\pi
i}\oint dz\;z^{k+\frac{1}{2}}\;S_{A}(z).\\\
&\overline{S}^{A}(\bar{z})=\sum_{l\in\mathbb{Z}+\frac{1}{2}}\frac{\overline{S}^{A}_{l}}{z^{l+\frac{3}{2}}},\quad\text{with}\quad\overline{S}^{A}_{l}=\frac{1}{2\pi
i}\oint
d\bar{z}\;\bar{z}^{l+\frac{1}{2}}\;\overline{S}^{A}(\bar{z})\end{split}$ (104)

then we can write the commutator of these modes with the Virasoro generators
as:

$\begin{split}&\left[L_{n},(S_{A})_{m}\right]=\left(\frac{n}{2}-m\right)(S_{A})_{m+n},\quad\left[\bar{L}_{n},(S_{A})_{m}\right]=0\\\
&\left[L_{n},\overline{S}^{A}_{m}\right]=0,\quad\left[\bar{L}_{n},\overline{S}^{A}_{m}\right]=\left(\frac{n}{2}-m\right)\overline{S}^{A}_{m+n}.\end{split}$
(105)

The operator relation (60) gives the anticommutator

$\\{(S_{B})_{m},\overline{S}^{A}_{n}\\}=\delta^{A}_{B}P_{mn},\quad
m,n\in\mathbb{Z}+\frac{1}{2}.$ (106)

Let us now discuss the requirement of (anti)holomorphicity of the singularity
in (77). Suppose $(\widetilde{\mathcal{G}}^{A}_{B})^{C}_{D}(z,\bar{z})$ has
conformal weights777the scaling dimensions of
$(\widetilde{\mathcal{G}}^{A}_{B})^{C}_{D}(z,\bar{z})$ can be calculated from
those of $G_{AB},\overline{G}^{CD}$. It is $(a+b^{\prime}-2,a^{\prime}+b-2)$.
$(h,\bar{h})$. Then we can expand the current as

$(\widetilde{\mathcal{G}}^{A}_{B})^{C}_{D}(z,\bar{z})=\sum_{n,m\in\mathbb{Z}}\big{\\{}(\widetilde{\mathcal{G}}^{A}_{B})^{C}_{D}\big{\\}}_{mn}z^{-m-h}\bar{z}^{-n-\bar{h}},$
(107)

with

$\big{\\{}(\widetilde{\mathcal{G}}^{A}_{B})^{C}_{D}\big{\\}}_{mn}=\frac{1}{(2\pi
i)^{2}}\oint dz\oint d\bar{z}\leavevmode\nobreak\
z^{m+h-1}\bar{z}^{n+\bar{h}+1}(\widetilde{\mathcal{G}}^{A}_{B})^{C}_{D}(z,\bar{z}).$
(108)

Suppose we had an OPE of the form888exactly same argument works if we have
higher power singularities.

$(\widetilde{\mathcal{G}}^{A}_{B})^{C}_{D}(z,\bar{z})S_{C}(w)\sim\frac{1}{z-w}\frac{1}{\bar{z}-\bar{w}}(T^{A}_{B})^{C}_{D}S_{C}(w).$
(109)

One can readily check that this would give us the commutator

$\left[(\widetilde{\mathcal{G}}^{A}_{B})^{C}_{D}\big{\\}}_{mn},(S_{C})_{k}\right]=\bar{w}^{n}\left(T_{B}^{A}\right)_{D}^{C}\left\\{S_{C}\right\\}_{m+k}$
(110)

This is nonsensical since we do not have any $\bar{w}$ dependence on the left
hand side. Similarly one can justify the second part of (77). So we conclude
that the $\mathcal{N}=8\leavevmode\nobreak\ \mathfrak{sbms}_{4}$ algebra does
not contain the extension of global R-symmetry algebra. The final algebra is
then given by

$\begin{split}&\left[L_{m},L_{n}\right]=(m-n)L_{m+n},\quad\left[\bar{L}_{m},\bar{L}_{n}\right]=(m-n)\bar{L}_{m+n}\\\
&\left[L_{n},P_{kl}\right]=\left(\frac{1}{2}n-k\right)P_{n+k,l},\quad\left[\bar{L}_{n},P_{kl}\right]=\left(\frac{1}{2}n-l\right)P_{k,n+l},\\\
&\left[L_{n},(S_{A})_{m}\right]=\left(\frac{n}{2}-m\right)(S_{A})_{m+n},\quad\left[\bar{L}_{n},(S_{A})_{m}\right]=0\\\
&\left[L_{n},\overline{S}^{A}_{m}\right]=0,\quad\left[\bar{L}_{n},\overline{S}^{A}_{m}\right]=\left(\frac{n}{2}-m\right)\overline{S}^{A}_{m+n}\\\
&\\{(S_{B})_{m},\overline{S}^{A}_{n}\\}=\delta^{A}_{B}P_{mn}.\end{split}$
(111)

##  5 Conclusion

In this paper, we have used the CCFT technique to compute the asymptotic
symmetry algebra of $\mathcal{N}=8$ supergravity in asymptotically flat
spacetime. The crucial part of our result is the non-extension of the global
$\mathrm{SU}(8)_{R}$ R-symmetry algebra. The purely mathematical
considerations Banerjee:2022abf  for $\mathcal{N}=2$ theory suggests that the
infinite dimensional extension of R-symmetry is fraught with mathematical
inconsistencies. Here, performing a direct asymptotic symmetry analysis of the
supergravity theory using CCFT prescription, we have confirmed that indeed
supergravity does not result in such an extension. The rest of the symmetry
algebra is as expected and is presented in (111).

Let us end the paper with relevant open problems. In the seminal work of
Hawking et.al. Hawking:2016sgy  the importance of the infinite number of soft
hairs in the context of black hole microscopics was discussed. The study was
further taken forward in Haco:2018ske ; Hawking:2016msc ; Hawking:2016sgy ;
Chu:2018tzu ; Averin:2016hhm  and beautifully reviewed in Strominger:2017zoo .
They emphasized the importance of symmetry enhancements at the future horizon
$\mathcal{H}^{+}$ of the black holes and how both the
hypersurfaces999$\mathcal{I}^{+}$ denotes the future null horizon
$\mathcal{H}^{+}$ and $\mathcal{I}^{+}$ carry information of conserved
charges, that are in turn important for understanding black hole microscopics.
The study of the present paper indicates that the asymptotic soft hairs of the
supergravity theories will not have distinct infinite R-charges, rather they
will only carry the global fixed number of R-charges. An interesting question
that remains to be studied is the effect, if any, of these R-charges at the
horizon and finally their importance in the black hole microscopics. We hope
to return to this question in future.

## Acknowledgement

The authors would like to thank Shamik Banerjee, Arpita Mitra, Debanshu
Mukherjee and Ashoke Sen for useful discussions. NB and TR are thankful to
participants to Regional String Meeting-Bhopal chapter for sharing interesting
outlook relevant for the present study. RKS would like to thank Anindya
Banerjee for various useful discussions. The work of RKS is supported by the
US Department of Energy under grant DE-SC0010008. Finally we are thankful to
the people of India for their generous support to the study of basic sciences.

##  Appendix A OPEs of component fields

Using (18), we can extract rest of the OPEs from the from the collinear
singularities of the amplitudes calculated in tab . In the following, the
zero, one, two, three and four index operators are respectively graviton,
gravitino, graviphoton, graviphotino and scalar operators.

###  A.1 Same spin OPEs

$\begin{split}&\mathcal{O}^{A}_{\Delta_{1},+\frac{3}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{B}_{\Delta_{2},+\frac{3}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-\frac{1}{2},\Delta_{2}-\frac{1}{2}\right)\mathcal{O}^{AB}_{\Delta_{1}+\Delta_{2},+1}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}^{A}_{\Delta_{1},+\frac{3}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{B\;\Delta_{2},-\frac{3}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{\bar{z}_{12}}{z_{12}}\;\delta^{A}_{B}\;B\left(\Delta_{1}-\frac{1}{2},\Delta_{2}+\frac{5}{2}\right)\mathcal{O}_{\Delta_{1}+\Delta_{2},-2}\left(z_{2},\bar{z}_{2}\right)\\\
&\hskip
150.69397pt+\frac{z_{12}}{\bar{z}_{12}}\;\delta^{A}_{B}\;B\left(\Delta_{1}+\frac{5}{2},\Delta_{2}-\frac{1}{2}\right)\mathcal{O}_{\Delta_{1}+\Delta_{2},+2}\left(z_{2},\bar{z}_{2}\right)\end{split}$
(112)
$\begin{split}&\mathcal{O}^{AB}_{\Delta_{1},+1}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{CD}_{\Delta_{2},+1}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1},\Delta_{2}\right)\mathcal{O}^{ABCD}_{\Delta_{1}+\Delta_{2},0}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}^{AB}_{\Delta_{1},+1}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{CD;\;\Delta_{2},-1}\left(z_{2,}\bar{z}_{2}\right)\sim-\delta^{AB}_{CD}\Big{[}\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1},\Delta_{2}+2\right)\mathcal{O}_{\Delta_{1}+\Delta_{2},-2}\left(z_{2},\bar{z}_{2}\right)\\\
&\hskip
150.69397pt+\frac{z_{12}}{\bar{z}_{12}}B\left(\Delta_{1}+2,\Delta_{2}\right)\mathcal{O}_{\Delta_{1}+\Delta_{2},+2}\left(z_{2},\bar{z}_{2}\right)\Big{]}\end{split}$
(113)

In the following, the notation is $a,b,c,\dots\in\\{1,2,3,4\\}$ and
$r,s,t,\dots\in\\{5,6,7,8\\}$. See tab  for details.

$\begin{split}&\mathcal{O}^{ars}_{\Delta_{1},+\frac{1}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{btu}_{\Delta_{2},+\frac{1}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim\epsilon^{rstu}\epsilon^{abcd}\frac{z_{12}}{\bar{z}_{12}}B\left(\Delta_{1}+\frac{1}{2},\Delta_{2}+\frac{1}{2}\right)\mathcal{O}_{cd;\Delta_{1}+\Delta_{2},-1}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}^{ars}_{\Delta_{1},+\frac{1}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{bct}_{\Delta_{2},+\frac{1}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim\epsilon^{rstu}\epsilon^{abcd}\frac{z_{12}}{\bar{z}_{12}}B\left(\Delta_{1}+\frac{1}{2},\Delta_{2}+\frac{1}{2}\right)\mathcal{O}_{ud;\Delta_{1}+\Delta_{2},-1}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}^{rst}_{\Delta_{1},+\frac{1}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{abc}_{\Delta_{2},+\frac{1}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim\epsilon^{rstu}\epsilon^{abcd}\frac{z_{12}}{\bar{z}_{12}}B\left(\Delta_{1}+\frac{1}{2},\Delta_{2}+\frac{1}{2}\right)\mathcal{O}_{ud;\Delta_{1}+\Delta_{2},-1}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}^{ars}_{\Delta_{1},+\frac{1}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{btu;\Delta_{2},-\frac{1}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim\epsilon_{tuvw}\epsilon^{rsvw}\delta^{a}_{b}\Big{[}\frac{z_{12}}{\bar{z}_{12}}B\left(\Delta_{1}+\frac{3}{2},\Delta_{2}+\frac{1}{2}\right)\mathcal{O}_{\Delta_{1}+\Delta_{2},+2}\left(z_{2},\bar{z}_{2}\right)\\\
&\hskip
150.69397pt+\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}+\frac{1}{2},\Delta_{2}+\frac{3}{2}\right)\mathcal{O}_{\Delta_{1}+\Delta_{2},-2}\left(z_{2},\bar{z}_{2}\right)\Big{]}\end{split}$
(114)
$\begin{split}&\mathcal{O}^{abrs}_{\Delta_{1},0}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{cdtu}_{\Delta_{2},0}\left(z_{2,}\bar{z}_{2}\right)\sim\epsilon^{abcd}\epsilon^{rstu}B\left(\Delta_{1}+1,\Delta_{2}+1\right)\Big{[}\frac{z_{12}}{\bar{z}_{12}}\mathcal{O}_{\Delta_{1}+\Delta_{2},+2}\left(z_{2},\bar{z}_{2}\right)\\\
&\hskip
172.22168pt+\frac{\bar{z}_{12}}{z_{12}}\mathcal{O}_{\Delta_{1}+\Delta_{2},-2}\left(z_{2},\bar{z}_{2}\right)\Big{]}\end{split}$
(115)

###  A.2 Different spins

$\begin{split}&\mathcal{O}_{\Delta_{1},+2}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{A}_{\Delta_{2},+\frac{3}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-1,\Delta_{2}-\frac{1}{2}\right)\mathcal{O}^{A}_{\Delta_{1}+\Delta_{2},+\frac{3}{2}}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}_{\Delta_{1},+2}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{A;\Delta_{2},-\frac{3}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-1,\Delta_{2}+\frac{5}{2}\right)\mathcal{O}_{A;\Delta_{1}+\Delta_{2},-\frac{3}{2}}\left(z_{2},\bar{z}_{2}\right)\end{split}$
(116)
$\begin{split}&\mathcal{O}_{\Delta_{1},+2}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{AB}_{\Delta_{2},+1}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-1,\Delta_{2}\right)\mathcal{O}^{AB}_{\Delta_{1}+\Delta_{2},+1}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}_{\Delta_{1},+2}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{AB;\Delta_{2},-1}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-1,\Delta_{2}+2\right)\mathcal{O}_{AB;\;\Delta_{1}+\Delta_{2},-1}\left(z_{2},\bar{z}_{2}\right)\end{split}$
(117)
$\begin{split}&\mathcal{O}_{\Delta_{1},+2}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{abr}_{\Delta_{2},+\frac{1}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-1,\Delta_{2}+\frac{1}{2}\right)\mathcal{O}^{abr}_{\Delta_{1}+\Delta_{2},+\frac{1}{2}}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}_{\Delta_{1},+2}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{abc}_{\Delta_{2},+\frac{1}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-1,\Delta_{2}+\frac{1}{2}\right)\mathcal{O}^{abc}_{\Delta_{1}+\Delta_{2},+\frac{1}{2}}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}_{\Delta_{1},+2}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{abc;\Delta_{2},-\frac{1}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim-\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-1,\Delta_{2}+\frac{3}{2}\right)\mathcal{O}_{abc;\;\Delta_{1}+\Delta_{2},-\frac{1}{2}}\left(z_{2},\bar{z}_{2}\right)\end{split}$
(118)
$\begin{split}\mathcal{O}_{\Delta_{1},+2}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{ABCD}_{\Delta_{2},0}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-1,\Delta_{2}+1\right)\mathcal{O}^{ABCD}_{\;\Delta_{1}+\Delta_{2},0}\left(z_{2},\bar{z}_{2}\right)\end{split}$
(119)
$\begin{split}&\mathcal{O}^{A}_{\Delta_{1},+\frac{3}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{BC}_{\Delta_{2},+1}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-\frac{1}{2},\Delta_{2}\right)\mathcal{O}^{ABC}_{\;\Delta_{1}+\Delta_{2},+\frac{1}{2}}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}^{A}_{\Delta_{1},+\frac{3}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{BC;\;\Delta_{2},-1}\left(z_{2,}\bar{z}_{2}\right)\sim
2!\;\delta^{A}_{[B}\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-\frac{1}{2},\Delta_{2}\right)\mathcal{O}_{C];\;\Delta_{1}+\Delta_{2},-\frac{3}{2}}\left(z_{2},\bar{z}_{2}\right)\end{split}$
(120)
$\begin{split}&\mathcal{O}^{A}_{\Delta_{1},+\frac{3}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{BCD}_{\Delta_{2},+\frac{1}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-\frac{1}{2},\Delta_{2}+\frac{1}{2}\right)\mathcal{O}^{ABCD}_{\;\Delta_{1}+\Delta_{2},0}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}^{A}_{\Delta_{1},+\frac{3}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{BCD\Delta_{2},-\frac{1}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim
3\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-\frac{1}{2},\Delta_{2}+\frac{3}{2}\right)\delta^{A}_{[B}\mathcal{O}_{CD];\;\Delta_{1}+\Delta_{2},-1}\left(z_{2},\bar{z}_{2}\right)\end{split}$
(121)
$\begin{split}&\mathcal{O}^{A}_{\Delta_{1},+\frac{3}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{BCDE}_{\Delta_{2},0}\left(z_{2},\bar{z}_{2}\right)\sim-\frac{1}{6}\epsilon^{ABCDEFGH}\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-\frac{1}{2},\Delta_{2}+1\right)\mathcal{O}_{FGH;\;\Delta_{1}+\Delta_{2},-\frac{1}{2}}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}^{A}_{\Delta_{1},+\frac{3}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{BCDE\Delta_{2},0}\left(z_{2},\bar{z}_{2}\right)\sim
3!\delta^{A}_{[B}\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}-\frac{1}{2},\Delta_{2}+1\right)\mathcal{O}_{CDE];\;\Delta_{1}+\Delta_{2},-\frac{1}{2}}\left(z_{2},\bar{z}_{2}\right)\end{split}$
(122)
$\begin{split}&\mathcal{O}^{ab}_{\Delta_{1},+1}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{cdr}_{\Delta_{2},+\frac{1}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim\frac{1}{3!}\epsilon^{rstu}\epsilon^{abcd}\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1},\Delta_{2}+\frac{1}{2}\right)\mathcal{O}_{stu;\;\Delta_{1}+\Delta_{2},-\frac{1}{2}}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}^{AB}_{\Delta_{1},+1}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{CDE;\;\Delta_{2},-\frac{1}{2}}\left(z_{2,}\bar{z}_{2}\right)\sim-\delta^{AB}_{CD}\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1},\Delta_{2}+\frac{3}{2}\right)\mathcal{O}_{E;\;\Delta_{1}+\Delta_{2},-\frac{3}{2}}\left(z_{2},\bar{z}_{2}\right)\end{split}$
(123)
$\begin{split}&\mathcal{O}^{ab}_{\Delta_{1},+1}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{cdrs}_{\Delta_{2},0}\left(z_{2,}\bar{z}_{2}\right)\sim\epsilon^{rstu}\epsilon^{abcd}\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1},\Delta_{2}+1\right)\mathcal{O}_{tu;\;\Delta_{1}+\Delta_{2},-1}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}^{ab}_{\Delta_{1},+1}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{cdef}_{\Delta_{2},0}\left(z_{2,}\bar{z}_{2}\right)\sim-\epsilon^{cdef}\epsilon^{abgh}\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1},\Delta_{2}+1\right)\mathcal{O}_{gh;\;\Delta_{1}+\Delta_{2},-1}\left(z_{2},\bar{z}_{2}\right)\end{split}$
(124)
$\begin{split}&\mathcal{O}^{abr}_{\Delta_{1},+\frac{1}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{cdst}_{\Delta_{2},0}\left(z_{2,}\bar{z}_{2}\right)\sim\epsilon^{rstu}\epsilon^{abcd}\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}+\frac{1}{2},\Delta_{2}+1\right)\mathcal{O}_{u;\;\Delta_{1}+\Delta_{2},-\frac{3}{2}}\left(z_{2},\bar{z}_{2}\right)\\\
&\mathcal{O}^{abr}_{\Delta_{1},+\frac{1}{2}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{cstu}_{\Delta_{2},0}\left(z_{2,}\bar{z}_{2}\right)\sim-\epsilon^{rstu}\epsilon^{abcd}\frac{\bar{z}_{12}}{z_{12}}B\left(\Delta_{1}+\frac{1}{2},\Delta_{2}+1\right)\mathcal{O}_{d;\;\Delta_{1}+\Delta_{2},-\frac{3}{2}}\left(z_{2},\bar{z}_{2}\right)\end{split}$
(125)

Similarly all other OPEs can be extracted from the amplitudes given in tab .

##  Appendix B OPE of the composite current $\mathcal{G}^{CD}_{AB}(z,\bar{z})$

We begin by calculating the OPEs $G\overline{G}$. We have101010Since there is
also an overlap as there in case of soft gravitino currents in section 3.1 we
do not seperate the operators in the correlator according to their spins
$\ell,\ell^{{}^{\prime}}$ and keep the spins to be arbitrary here as well.

$\begin{split}&\left\langle
G_{AB}(z,\bar{z})\overline{G}^{CD}(w,\bar{w})\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=\lim_{\begin{subarray}{c}\Delta_{1}\to 0\\\ \Delta_{2}\to
0\end{subarray}}\frac{\Delta_{1}\Delta_{2}}{\pi^{2}}\int
d^{2}z_{1}\frac{1}{\left(z-z_{1}\right)^{a}}\frac{1}{\left(\bar{z}-\bar{z}_{1}\right)^{b}}\frac{1}{\left(\bar{w}-\bar{z}_{2}\right)^{a^{\prime}}}\frac{1}{\left(w-z_{2}\right)^{b^{\prime}}}\\\
&\hskip 56.9055pt\times\left\langle\mathcal{O}_{AB\leavevmode\nobreak\
\Delta_{1},-1}(z_{1},\bar{z}_{1})\mathcal{O}^{CD}_{\Delta_{2},+1}(z_{2},\bar{z}_{2})\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\end{split}$
(126)

By taking the soft limit of the first operator $\Delta_{1}\to 0$,

$\begin{split}&\left\langle
G_{AB}(z)\overline{G}^{CD}(\bar{w})\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=\lim_{\begin{subarray}{c}\Delta_{2}\to
0\end{subarray}}\frac{\Delta_{2}}{\pi^{2}}\int d^{2}z_{1}\int
d^{2}z_{2}\frac{1}{\left(z-z_{1}\right)^{a}}\frac{1}{\left(\bar{z}-\bar{z}_{1}\right)^{b}}\frac{1}{\left(\bar{w}-\bar{z}_{2}\right)^{a^{\prime}}}\frac{1}{\left(w-z_{2}\right)^{b^{\prime}}}\\\
&\times\Bigg{[}-\delta^{CD}_{AB}\frac{z_{1}-z_{2}}{\bar{z}_{1}-\bar{z}_{2}}\left\langle\mathcal{O}_{\Delta_{2},+2}(z_{2},\bar{z}_{2})\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&+\sum_{j=3}^{N}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})\frac{z_{1}-z_{j}}{\bar{z}_{1}-\bar{z}_{j}}\left\langle\mathcal{O}^{CD}_{\Delta_{2},+1}(z_{2},\bar{z}_{2})\cdots\mathcal{O}^{*^{\prime}_{j}}_{\Delta_{j},\ell_{j}+1}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\Bigg{]}\end{split}$
(127)

Now doing the first integral using (80), we get

$\begin{split}&\left\langle
G_{AB}(z)\overline{G}^{CD}(\bar{w})\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=\lim_{\begin{subarray}{c}\Delta_{2}\to
0\end{subarray}}\frac{\Delta_{2}}{\pi}\int
d^{2}z_{2}\frac{1}{\left(\bar{w}-\bar{z}_{2}\right)^{a^{\prime}}}\frac{1}{\left(w-z_{2}\right)^{b^{\prime}}}\\\
&\times\Bigg{[}-\delta^{CD}_{AB}C_{1}(b,a)\frac{1}{\left(z_{2}-z\right)^{a-2}\left(\bar{z}_{2}-\bar{z}\right)^{b}}\left\langle\mathcal{O}_{\Delta_{2},+2}(z_{2},\bar{z}_{2})\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&+\sum_{j=3}^{N}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})C_{1}(b,a)\frac{1}{\left(z_{j}-z\right)^{a-2}}\frac{1}{\left(\bar{z}_{j}-\bar{z}\right)^{b}}\\\
&\hskip
142.26378pt\times\left\langle\mathcal{O}^{CD}_{\Delta_{2},+1}(z_{2},\bar{z}_{2})\cdots\mathcal{O}^{*^{\prime}_{j}}_{\Delta_{j},\ell_{j}+1}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\Bigg{]}\end{split}$
(128)

We now use the collinear limits of the graviton operator with other fields in
the first term and take the conformally soft limit $\Delta_{2}\to 0$ in the
second term.
The first term becomes

$\begin{split}&\lim_{\begin{subarray}{c}\Delta_{2}\to
0\end{subarray}}\frac{\Delta_{2}}{\pi}\int
d^{2}z_{2}\frac{1}{\left(\bar{w}-\bar{z}_{2}\right)^{a^{\prime}}}\frac{1}{\left(w-z_{2}\right)^{b^{\prime}}}\Bigg{[}-\delta^{CD}_{AB}C_{1}(b,a)\frac{1}{\left(z_{2}-z\right)^{a-2}\left(\bar{z}_{2}-\bar{z}\right)^{b}}\\\
&\hskip
85.35826pt\times\left\langle\mathcal{O}_{\Delta_{2},+2}(z_{2},\bar{z}_{2})\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\Bigg{]}\\\
&=-\delta^{CD}_{AB}C_{1}(b,a)\lim_{\begin{subarray}{c}\Delta_{2}\to
0\end{subarray}}\frac{\Delta_{2}}{\pi}\int
d^{2}z_{2}\frac{1}{\left(\bar{w}-\bar{z}_{2}\right)^{a^{\prime}}}\frac{1}{\left(w-z_{2}\right)^{b^{\prime}}}\frac{1}{\left(z_{2}-z\right)^{a-2}\left(\bar{z}_{2}-\bar{z}\right)^{b}}\\\
&\times\sum_{i=3}^{N}B(\Delta_{2}-1,f(\Delta_{i}))\frac{\bar{z}_{2}-\bar{z}_{i}}{z_{2}-z_{i}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*_{i}}_{\Delta_{i},\ell_{i}}(z_{i},\bar{z}_{i})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&=-\delta^{CD}_{AB}C_{1}(b,a)\frac{1}{\pi}\int
d^{2}z_{2}\frac{1}{\left(\bar{w}-\bar{z}_{2}\right)^{a^{\prime}}}\frac{1}{\left(w-z_{2}\right)^{b^{\prime}}}\frac{1}{\left(z_{2}-z\right)^{a-2}\left(\bar{z}_{2}-\bar{z}\right)^{b}}\\\
&\times\sum_{i=3}^{N}(1-f(\Delta_{i}))\frac{\bar{z}_{2}-\bar{z}_{i}}{z_{2}-z_{i}}\times\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*_{i}}_{\Delta_{i},\ell_{i}}(z_{i},\bar{z}_{i})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\end{split}$
(129)

where we used

$\begin{split}\lim_{\begin{subarray}{c}\Delta_{2}\to
0\end{subarray}}\Delta_{2}\;B(\Delta_{2}-1,f(\Delta_{i}))=1-f(\Delta_{i})\end{split}$

In the second term in (128) now we can take $\Delta_{2}\to 0$ limit,

$\begin{split}&\lim_{\begin{subarray}{c}\Delta_{2}\to
0\end{subarray}}\frac{\Delta_{2}}{\pi}\sum_{j=3}^{N}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})\frac{1}{\left(z_{j}-z\right)^{a-2}}\frac{1}{\left(\bar{z}_{j}-\bar{z}\right)^{b}}\int
d^{2}z_{2}\frac{1}{\left(\bar{w}-\bar{z}_{2}\right)^{a^{\prime}}}\frac{1}{\left(w-z_{2}\right)^{b^{\prime}}}\\\
&\hskip
142.26378pt\times\left\langle\mathcal{O}^{CD}_{\Delta_{2},+1}(z_{2},\bar{z}_{2})\cdots\mathcal{O}^{*^{\prime}_{j}}_{\Delta_{j},\ell_{j}+1}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&=\frac{1}{\pi}\sum_{j=3}^{N}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})\bar{f}(C,D,\ell_{j}+1,*_{j}^{\prime},*_{j}^{\prime\prime})C_{1}(b,a)\frac{1}{\left(z_{j}-z\right)^{a-2}}\frac{1}{\left(\bar{z}_{j}-\bar{z}\right)^{b}}\\\
&\times\int
d^{2}z_{2}\frac{1}{\left(\bar{w}-\bar{z}_{2}\right)^{a^{\prime}}}\frac{1}{\left(w-z_{2}\right)^{b^{\prime}}}\frac{\bar{z}_{2}-\bar{z}_{j}}{z_{2}-z_{j}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*^{\prime\prime}_{j}}_{\Delta_{j},\ell_{j}}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&+\frac{1}{\pi}\sum_{\begin{subarray}{c}i,j=3\\\ i\neq
j\end{subarray}}^{N}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})\bar{f}(C,D,\ell_{i},*_{i},*_{i}^{\prime})C_{1}(b,a)\frac{1}{\left(z_{j}-z\right)^{a-2}}\frac{1}{\left(\bar{z}_{j}-\bar{z}\right)^{b}}\\\
&\times\int
d^{2}z_{2}\frac{1}{\left(\bar{w}-\bar{z}_{2}\right)^{a^{\prime}}}\frac{1}{\left(w-z_{2}\right)^{b^{\prime}}}\frac{\bar{z}_{2}-\bar{z}_{i}}{z_{2}-z_{i}}\left\langle\cdots\mathcal{O}^{*^{\prime}_{j}}_{\Delta_{j},\ell_{j}+1}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*^{\prime}_{i}}_{\Delta_{i},\ell_{i}-1}(z_{i},\bar{z}_{i})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&=\sum_{j=3}^{N}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})\bar{f}(C,D,\ell_{j}+1,*_{j}^{\prime},*_{j}^{\prime\prime})C_{1}(b^{\prime},a^{\prime})\frac{1}{\left(z_{j}-z\right)^{a-2}}\frac{1}{\left(\bar{z}_{j}-\bar{z}\right)^{b}}\\\
&\times\frac{1}{\left(z_{j}-w\right)^{b^{\prime}}}\frac{1}{\left(\bar{z}_{j}-\bar{w}\right)^{a^{\prime}-2}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*^{{}^{\prime\prime}}_{j}}_{\Delta_{j},\ell_{j}}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&+\sum_{\begin{subarray}{c}i,j=3\\\ i\neq
j\end{subarray}}^{N}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})\bar{f}(C,D,\ell_{i},*_{i},*_{i}^{\prime})C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})\frac{1}{\left(z_{j}-z\right)^{a-2}}\frac{1}{\left(\bar{z}_{j}-\bar{z}\right)^{b}}\\\
&\times\frac{1}{\left(z_{i}-w\right)^{b^{\prime}}}\frac{1}{\left(\bar{z}_{i}-\bar{w}\right)^{a^{\prime}-2}}\left\langle\cdots\mathcal{O}^{*^{\prime}_{j}}_{\Delta_{j},\ell_{j}+1}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*^{\prime}_{i}}_{\Delta_{i},\ell_{i}-1}(z_{i},\bar{z}_{i})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\end{split}$
(130)

Combining the two integrals we get

$\begin{split}&\left\langle
G_{AB}(z,\bar{z})\overline{G}^{CD}(w,\bar{w})\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=-\delta^{CD}_{AB}C_{1}(b,a)\frac{1}{\pi}\int
d^{2}z_{2}\frac{1}{\left(\bar{w}-\bar{z}_{2}\right)^{a^{\prime}}}\frac{1}{\left(w-z_{2}\right)^{b^{\prime}}}\frac{1}{\left(z_{2}-z\right)^{a-2}\left(\bar{z}_{2}-\bar{z}\right)^{b}}\\\
&\times\sum_{i=3}^{N}(1-f(\Delta_{i}))\frac{\bar{z}_{2}-\bar{z}_{i}}{z_{2}-z_{i}}\times\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*_{i}}_{\Delta_{i},\ell_{i}}(z_{i},\bar{z}_{i})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&+\sum_{j=3}^{N}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})\bar{f}(C,D,\ell_{j}+1,*_{j}^{\prime},*_{j}^{\prime\prime})C_{1}(b^{\prime},a^{\prime})\frac{1}{\left(z_{j}-z\right)^{a-2}}\frac{1}{\left(\bar{z}_{j}-\bar{z}\right)^{b}}\\\
&\times\frac{1}{\left(z_{j}-w\right)^{b^{\prime}}}\frac{1}{\left(\bar{z}_{j}-\bar{w}\right)^{a^{\prime}-2}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*^{{}^{\prime\prime}}_{j}}_{\Delta_{j},\ell_{j}}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&+\sum_{\begin{subarray}{c}i,j=3\\\ i\neq
j\end{subarray}}^{N}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})\bar{f}(C,D,\ell_{i},*_{i},*_{i}^{\prime})C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})\frac{1}{\left(z_{j}-z\right)^{a-2}}\frac{1}{\left(\bar{z}_{j}-\bar{z}\right)^{b}}\\\
&\times\frac{1}{\left(z_{i}-w\right)^{b^{\prime}}}\frac{1}{\left(\bar{z}_{i}-\bar{w}\right)^{a^{\prime}-2}}\left\langle\cdots\mathcal{O}^{*^{\prime}_{j}}_{\Delta_{j},\ell_{j}+1}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*^{\prime}_{i}}_{\Delta_{i},\ell_{i}-1}(z_{i},\bar{z}_{i})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\end{split}$
(131)

Here when we take the normal order of this composite current we only need to
care about the non-singular terms in the above OPE. The non-singular term in
the integral above can be obtained by taking $z\to w$ limit in the integral.
The integral can then be evaluated as

$\displaystyle\int
d^{2}z_{2}\frac{1}{\left(\bar{w}-\bar{z}_{2}\right)^{a^{\prime}}}$
$\displaystyle\frac{1}{\left(w-z_{2}\right)^{b^{\prime}}}\frac{1}{\left(z_{2}-z\right)^{a-2}}\frac{1}{\left(\bar{z}_{2}-\bar{z}\right)^{b}}\frac{\bar{z}_{2}-\bar{z}_{i}}{z_{2}-z_{i}}\quad\stackrel{{\scriptstyle
z=w}}{{\longrightarrow}}$ $\displaystyle=(-1)^{a+b-2}\int
d^{2}z_{2}\frac{1}{\left(\bar{z}-\bar{z}_{2}\right)^{a^{\prime}+b}}\frac{1}{\left(z-z_{2}\right)^{b^{\prime}+a-2}}\frac{\bar{z}_{2}-\bar{z}_{i}}{z_{2}-z_{i}}$
$\displaystyle=(-1)^{a+b}C_{1}\left(a+b^{\prime}-2,a^{\prime}+b\right)\frac{1}{\left(z_{i}-z\right)^{a^{\prime}+b-2}}\frac{1}{\left(\bar{z}_{i}-\bar{z}\right)^{a+b^{\prime}-2}}$

Hence

$\begin{split}&\left\langle:G_{AB}(z,\bar{z})\overline{G}^{CD}(z,\bar{z}):\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=\frac{(-1)^{a+b+1}}{\pi}\delta^{CD}_{AB}C_{1}(b,a)C\left(a+b^{\prime}-2,a^{\prime}+b\right)\sum_{i=3}^{N}(1-f(\Delta_{i}))\\\
&\hskip
28.45274pt\times\frac{1}{\left(z_{i}-z\right)^{a^{\prime}+b-2}}\frac{1}{\left(\bar{z}_{i}-\bar{z}\right)^{a+b^{\prime}-2}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*_{i}}_{\Delta_{i},\ell_{i}}(z_{i},\bar{z}_{i})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell^{\prime}_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&+\sum_{j=3}^{N}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})\bar{f}(C,D,\ell_{j}+1,*_{j}^{\prime},*_{j}^{\prime\prime})C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})(-1)^{a+b+a^{\prime}+b^{\prime}}\\\
&\hskip
28.45274pt\times\frac{1}{\left(z-z_{j}\right)^{a+b^{\prime}-2}}\frac{1}{\left(\bar{z}-\bar{z}_{j}\right)^{a^{\prime}+b-2}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*^{{}^{\prime\prime}}_{j}}_{\Delta_{j},\ell_{j}}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&+\sum_{\begin{subarray}{c}i,j=3\\\ i\neq
j\end{subarray}}^{N}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})\bar{f}(C,D,\ell_{i},*_{i},*_{i}^{\prime})C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})(-1)^{a+b+a^{\prime}+b^{\prime}}\frac{1}{\left(z-z_{j}\right)^{a-2}}\frac{1}{\left(\bar{z}-\bar{z}_{j}\right)^{b}}\\\
&\hskip
28.45274pt\times\frac{1}{\left(z-z_{i}\right)^{b^{\prime}}}\frac{1}{\left(\bar{z}-\bar{z}_{i}\right)^{a^{\prime}-2}}\left\langle\cdots\mathcal{O}^{*^{\prime}_{j}}_{\Delta_{j},\ell_{j}+1}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*^{\prime}_{i}}_{\Delta_{i},\ell_{i}-1}(z_{i},\bar{z}_{i})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\end{split}$
(132)

Similarly we have,

$\begin{split}&\left\langle:\overline{G}^{CD}(z,\bar{z})G_{AB}(z,\bar{z}):\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=\frac{(-1)^{a^{\prime}+b^{\prime}+1}}{\pi}\delta^{CD}_{AB}C_{1}(b^{\prime},a^{\prime})C\left(a^{\prime}+b-2,a+b^{\prime}\right)\sum_{i=3}^{N}(1-f(\Delta_{i}))\\\
&\hskip
28.45274pt\times\frac{1}{\left(z_{i}-z\right)^{a+b^{\prime}-2}}\frac{1}{\left(\bar{z}_{i}-\bar{z}\right)^{a^{\prime}+b-2}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*_{i}}_{\Delta_{i},\ell_{i}}(z_{i},\bar{z}_{i})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&+\sum_{j=3}^{N}\bar{f}(C,D,\ell_{j},*_{j},*_{j}^{\prime})f(A,B,\ell_{j}-1,*_{j}^{\prime},*_{j}^{\prime\prime})C_{1}(b^{\prime},a^{\prime})C_{1}(b,a)(-1)^{a+b+a^{\prime}+b^{\prime}}\\\
&\hskip
28.45274pt\times\frac{1}{\left(z-z_{j}\right)^{a^{\prime}+b-2}}\frac{1}{\left(\bar{z}-\bar{z}_{j}\right)^{a+b^{\prime}-2}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*^{{}^{\prime\prime}}_{j}}_{\Delta_{j},\ell_{j}}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&+\sum_{\begin{subarray}{c}i,j=3\\\ i\neq
j\end{subarray}}^{N}\bar{f}(C,D,\ell_{j},*_{j},*_{j}^{\prime})f(A,B,\ell_{i},*_{i},*_{i}^{\prime})C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})(-1)^{a+b+a^{\prime}+b^{\prime}}\frac{1}{\left(\bar{z}-\bar{z}_{j}\right)^{a^{\prime}-2}}\frac{1}{\left(z-z_{j}\right)^{b^{\prime}}}\\\
&\hskip
28.45274pt\times\frac{1}{\left(\bar{z}-\bar{z}_{i}\right)^{b}}\frac{1}{\left(z-z_{i}\right)^{a-2}}\left\langle\cdots\mathcal{O}^{*^{\prime}_{j}}_{\Delta_{j},\ell_{j}-1}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*^{\prime}_{i}}_{\Delta_{i},\ell_{i}+1}(z_{i},\bar{z}_{i})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\end{split}$
(133)

We have the correlator of the normalised current with any conformal primary
as,

$\begin{split}&\left\langle\mathcal{G}_{AB}^{CD}(z,\bar{z})\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=-\delta^{CD}_{AB}\sum_{i=3}^{N}(1-f(\Delta_{i}))\Bigg{[}\frac{(-1)^{a+b}}{\pi}C_{1}(b,a)C\left(a+b^{\prime}-2,a^{\prime}+b\right)\frac{1}{\left(z_{i}-z\right)^{a^{\prime}+b-2}}\frac{1}{\left(\bar{z}_{i}-\bar{z}\right)^{a+b^{\prime}-2}}\\\
&\hskip
142.26378pt\times\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*_{i}}_{\Delta_{i},\ell_{i}}(z_{i},\bar{z}_{i})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&-\frac{(-1)^{a^{\prime}+b^{\prime}}}{\pi}C_{1}(b^{\prime},a^{\prime})C_{1}\left(a^{\prime}+b-2,a+b^{\prime}\right)\frac{1}{\left(z_{i}-z\right)^{a+b^{\prime}-2}}\frac{1}{\left(\bar{z}_{i}-\bar{z}\right)^{a^{\prime}+b-2}}\\\
&\hskip
142.26378pt\times\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*_{i}}_{\Delta_{i},\ell_{i}}(z_{i},\bar{z}_{i})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\rangle\Bigg{]}\\\
&+(-1)^{a+b+a^{\prime}+b^{\prime}}C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})\Bigg{[}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})\bar{f}(C,D,\ell_{j}+1,*_{j}^{\prime},*_{j}^{\prime\prime})\\\
&\hskip
28.45274pt\times\frac{1}{\left(z-z_{j}\right)^{a+b^{\prime}-2}}\frac{1}{\left(\bar{z}-\bar{z}_{j}\right)^{a^{\prime}+b-2}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*^{{}^{\prime\prime}}_{j}}_{\Delta_{j},\ell_{j}}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&\hskip
56.9055pt-\bar{f}(C,D,\ell_{j},*_{j},*_{j}^{\prime})f(A,B,\ell_{j}-1,*_{j}^{\prime},*_{j}^{\prime\prime})\\\
&\hskip
28.45274pt\times\frac{1}{\left(z-z_{j}\right)^{a^{\prime}+b-2}}\frac{1}{\left(\bar{z}-\bar{z}_{j}\right)^{a+b^{\prime}-2}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*^{{}^{\prime\prime}}_{j}}_{\Delta_{j},\ell_{j}}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\Bigg{]}\end{split}$
(134)

The last term in both (132) and (133) cancels when we take the commutator. Now
in the above OPE we can see that the first term which has the graviton soft
limits, does not satisfy our requirement explained in (77). Hence we require
that the first terms in the two ordering to be the same, so that they cancel
once we take the commutator. This is equivalent to the requirement

$\displaystyle
C_{1}\left(b^{\prime},a^{\prime}\right)C_{1}\left(a^{\prime}+b-2,a+b^{\prime}\right)(-1)^{a+b}$
$\displaystyle=C_{1}(b,a)C_{1}\left(a+b^{\prime}-2,a^{\prime}+b\right)(-1)^{a^{\prime}+b^{\prime}}$
(135)

and

$\begin{split}a^{\prime}+b-2&=a+b^{\prime}-2\end{split}$ (136)

Now in (135) by substituting the explicit expression from (81) we have

$\left(-a^{\prime}-b+1\right)\left(-a^{\prime}-b+2\right)(-a+1)(-a+2)=\left(-a^{\prime}+1\right)\left(-a^{\prime}+2\right)\left(-a-b^{\prime}+1\right)\left(-a-b^{\prime}+2\right).$

which after using (136) gives

$(-a+1)(-a+2)=\left(-a^{\prime}+1\right)\left(-a^{\prime}+2\right)$ (137)

which clearly has solutions. Hence correlator corresponding to this normal
order current is,

$\begin{split}&\left\langle\mathcal{G}_{AB}^{CD}(z,\bar{z})\prod_{n=3}^{N}\mathcal{O}^{*_{n}}_{\Delta_{n},\ell_{n}}(z_{n},\bar{z}_{n})\right\rangle\\\
&=(-1)^{a+b+a^{\prime}+b^{\prime}}C_{1}(b,a)C_{1}(b^{\prime},a^{\prime})\Bigg{[}f(A,B,\ell_{j},*_{j},*_{j}^{\prime})\bar{f}(C,D,\ell_{j}+1,*_{j}^{\prime},*_{j}^{\prime\prime})\\\
&\hskip
28.45274pt\times\frac{1}{\left(z-z_{j}\right)^{a+b^{\prime}-2}}\frac{1}{\left(\bar{z}-\bar{z}_{j}\right)^{a^{\prime}+b-2}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*^{{}^{\prime\prime}}_{j}}_{\Delta_{j},\ell_{j}}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\\\
&\hskip
56.9055pt-\bar{f}(C,D,\ell_{j},*_{j},*_{j}^{\prime})f(A,B,\ell_{j}-1,*_{j}^{\prime},*_{j}^{\prime\prime})\\\
&\hskip
28.45274pt\times\frac{1}{\left(z-z_{j}\right)^{a^{\prime}+b-2}}\frac{1}{\left(\bar{z}-\bar{z}_{j}\right)^{a+b^{\prime}-2}}\left\langle\mathcal{O}^{*_{3}}_{\Delta_{3},\ell_{3}}(z_{3},\bar{z}_{3})\cdots\mathcal{O}^{*^{{}^{\prime\prime}}_{j}}_{\Delta_{j},\ell_{j}}(z_{j},\bar{z}_{j})\cdots\mathcal{O}^{*_{N}}_{\Delta_{N},\ell_{N}}(z_{N},\bar{z}_{N})\right\rangle\Bigg{]}\end{split}$
(138)

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