11institutetext: Institute of Physics, Sachivalaya Marg, Bhubaneshwar,
India-751005
and Homi Bhabha National Institute, Anushakti Nagar, Mumbai,
India-40008522institutetext: Okinawa Institute of Science and
Technology,1919-1 Tancha, Onna-son, Okinawa 904-0495,Japan33institutetext:
Chennai Mathematical Institute, SIPCOT IT Park, Siruseri 603103, India

# (Chiral) Virasoro invariance of the tree-level MHV graviton scattering
amplitudes

Shamik Banerjee 2 Sudip Ghosh 3 Partha Paul<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we continue our study of the tree level MHV graviton scattering
amplitudes from the point of view of celestial holography. In arXiv:2008.04330
we showed that the celestial OPE of two gravitons in the MHV sector can be
written as a linear combination of $\overline{SL(2,\mathbb{C})}$ current
algebra and supertranslation descendants. In this note we show that the OPE is
in fact manifestly invariant under the infinite dimensional Virasoro algebra
as is expected for a $2$-D CFT. This is consistent with the conjecture that
the holographic dual in $4$-D asymptotically flat space time is a $2$-D CFT.
Since we get only one copy of the Virasoro algebra we can conclude that the
holographic dual theory which computes the MHV amplitudes is a chiral CFT with
a host of other infinite dimensional global symmetries including
$\overline{SL(2,\mathbb{C})}$ current algebra, supertranslations and
subsubleading soft graviton symmetry. We also discuss some puzzles related to
the appearance of the Virasoro symmetry.

##  1 Introduction and Results

In four dimensional asymptotically flat space time the asymptotic symmetries
are infinite dimensional Sachs:1962zza ; Strominger:2013jfa ; He ;
Barnich:2009se ; Barnich:2011ct ; Kapec:2016jld ; Kapec:2014opa ; He:2017fsb ;
Banerjee:2020zlg ; Guevara:2021abz ; Banerjee:2021cly ; Strominger:2021lvk ;
Donnay:2020guq ; Bagchi:2016bcd  and so in a holographic description the dual
theory must have infinite dimensional global symmetry. It has been conjectured
Barnich:2009se ; Barnich:2011ct ; Kapec:2016jld ; Kapec:2014opa ; He:2017fsb ;
Bagchi:2016bcd  that the global symmetry group of the dual theory contains the
local conformal or Virasoro symmetry. As a result the dual theory is a two
dimensional CFT. In this note we study this conjecture in the context of
celestial holography Pasterski:2017kqt .

We revisit the global symmetries of the tree level MHV graviton scattering
amplitudes calculated in General Relativity. In Banerjee:2020zlg  we studied
the celestial MHV amplitudes in great detail and showed that the celestial OPE
between two graviton primaries in the MHV sector can be written as a linear
combination of $\overline{SL(2,\mathbb{C})}$ current algebra, supetranslations
and global $SL(2,\mathbb{C})$ descendants. Moreover, due to the existence of
the null states Banerjee:2020zlg 111The generators of the
$\overline{SL(2,\mathbb{C})}$ current algebra are denoted by $J^{a}_{n}$ where
$a=0,\pm 1$ and $n\in\mathbb{Z}$. Similarly we denote the generators of the
supertranslations coming from the positive helicity soft graviton by $P_{n,b}$
where $n\in\mathbb{Z}$ and $b=0,-1$. For further details on the symmetry
algebra and null states we refer the reader to Banerjee:2020zlg .,

$\Psi_{\Delta}(z,\bar{z})=\left[J^{1}_{-1}P_{-1,-1}-(\Delta-1)P_{-2,0}\right]G^{+}_{\Delta}(z,\bar{z})=0$
(1)

and

$\Phi_{\Delta}(z,\bar{z})=\left[L_{-1}P_{-1,-1}+2J^{0}_{-1}P_{-1,-1}-(\Delta+1)P_{-2,-1}-\bar{L}_{-1}P_{-2,0}\right]G^{+}_{\Delta}(z,\bar{z})=0$
(2)

one can write down differential equations which can be solved to determine the
MHV graviton scattering amplitudes. Therefore the MHV amplitudes are
completely determined by the $\overline{SL(2,\mathbb{C})}$ current algebra,
supetranslations and the global $SL(2,\mathbb{C})$ symmetries.

In this paper we point out that the celestial OPE of two graviotns in the MHV
sector is actually invariant under the infinite dimensional Viarasoro algebra
whose global part is the $SL(2,\mathbb{C})$ algebra. This shows that the
symmetries of the MHV graviton scattering amplitudes is a semi-direct product
of the Virasoro algebra, $\overline{SL(2,\mathbb{C})}$ current algebra and
supertranslations.222See paul  for a potential derivation of this symmetry
algebra from the bulk gravity point of view. Since we get only one copy of the
Virasoro algebra, the holographic dual theory which computes the MHV graviton
scattering amplitudes is a chiral (celestial) CFT.

For the sake of completeness let us now write down the symmetry algebra,

###  1.1 Virasoro algebra

$\left[L_{m},L_{n}\right]=(m-n)L_{m+n}+{\color[rgb]{1,0,0}\frac{c}{12}n(n^{2}-1)\delta_{m+n,0}}\quad
m,n\in\mathbb{Z}$ (3)

Although we have written down the central charge term in the Virasoro algebra,
our demonstration of Virasoro invariance of the celestial OPE does not allow
us to determine the value of the central charge $c$. So we leave the
determination of $c$ to future work.

###  1.2 $\overline{SL(2,\mathbb{C})}$ current algebra

$\left[J^{a}_{m},J^{b}_{n}\right]=(a-b)J^{a+b}_{m+n},\quad a,b=0,\pm 1,\quad
m,n\in\mathbb{Z}$ (4)

$J^{1}_{0}=\bar{L}_{1},\quad J^{0}_{0}=\bar{L}_{0},\quad
J^{-1}_{0}=\bar{L}_{-1}$ (5)

###  1.3 Supertranslation

$\left[P_{m,n},P_{m^{\prime},n^{\prime}}\right]=0,\quad n,n^{\prime}=0,-1$ (6)

###  1.4 Mixed commutators

$\quad[J^{1}_{m},P_{n,-1}]=P_{m+n,0},\quad[J^{0}_{m},P_{n,-1}]=\frac{1}{2}P_{m+n,-1},\quad[J^{-1}_{m},P_{n,-1}]=0$
(7)

$\quad\left[J^{1}_{m},P_{n,0}\right]=0,\quad\left[J^{0}_{m},P_{n,0}\right]=-\frac{1}{2}P_{m+n,0},\quad[J^{-1}_{m},P_{n,0}]=-P_{m+n,-1}$
(8)

$[L_{m},J^{a}_{n}]=-nJ^{a}_{m+n},\quad m\in\mathbb{Z},\quad n\in\mathbb{Z}$
(9)

$\left[L_{n},P_{a,b}\right]=\left(\frac{n-1}{2}-a\right)P_{a+n,b},\quad
n\in\mathbb{Z},\quad a\in\mathbb{Z},\quad b=0,-1$ (10)

The symmetry algebra that we have shown can be further extended by including
the subsubleading soft graviton symmetry Guevara:2021abz ; Banerjee:2021cly ;
Strominger:2021lvk  and an infinite number of other soft symmetries
Guevara:2021abz ; Strominger:2021lvk  which appear when the scaling dimension
of a (positive helicity) graviton primary assumes negative integer values.

##  2 Virasoro Invariance of the OPE in the MHV sector

Consider the celestial OPE between two positive helicity outgoing gravitons in
the MHV sector. This is given by Banerjee:2020zlg ,

$\begin{split}&G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)\Big{|}_{MHV}\\\
&=\frac{\bar{z}}{z}\
B(\Delta_{1}-1,\Delta_{2}-1)\bigg{[}-P_{-1,-1}+z\big{(}c_{1}\hskip
1.13791ptJ^{0}_{-1}P_{-1,-1}+c_{2}P_{-2,-1}\big{)}\\\
&+z^{2}\bigg{(}c_{3}\hskip
1.13791ptJ^{0}_{-2}P_{-1,-1}+c_{4}P_{-3,-1}+c_{5}\big{(}2L_{-1}P_{-2,-1}-2\bar{L}_{-1}P_{-3,0}+2L_{-1}\bar{L}_{-1}P_{-2,0}-L^{2}_{-1}P_{-1,-1}\big{)}\bigg{)}\\\
&+\mathcal{O}(z^{3})\bigg{]}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)+\cdots\end{split}$
(11)

The descendant OPE coefficients denoted above by $c_{i}$ are given by

$\begin{split}&c_{1}=\frac{2(\Delta_{1}-1)}{\Delta_{1}+\Delta_{2}-2},\quad
c_{2}=-\Delta_{1},\quad
c_{3}=\frac{2(\Delta_{1}-1)(\Delta_{2}-1)}{(\Delta_{1}+\Delta_{2}-2)(\Delta_{1}+\Delta_{2}-1)},\\\
&c_{4}=-\frac{\Delta_{1}\Delta_{2}}{\Delta_{1}+\Delta_{2}-1},\quad
c_{5}=\frac{\Delta_{1}(\Delta_{1}-1)}{2(\Delta_{1}+\Delta_{2}-2)(\Delta_{1}+\Delta_{2}-1)}\end{split}$
(12)

In (11) we have, for simplicity, kept only terms of the
$\mathcal{O}(\bar{z}z^{n})$ where $n\geq-1$ because under the action of the
Virasoro generators these terms do not mix with other terms in the OPE.

We already know Banerjee:2020zlg  that the OPE (11) is invariant under the
action of the global generator $L_{1}$. In this section we show that both
sides of the OPE (11) transform in the same way under the action of the first
Virasoro generator $L_{2}$. This check is non-trivial because the OPE (11) has
terms of $\mathcal{O}(\bar{z}z)$.

It is important to note that the $\mathcal{O}(\bar{z}z)$ term in (11) does not
contain the $L_{-2}$ descendant of the graviton primary and so the Virasoro
central charge $c$ does not play any role in the process of checking the
Virasoro invariance. Let us now spell out the details.

###  2.1 $L_{2}$ invariance

Let us start with the OPE

$\begin{split}&G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)\Big{|}_{MHV}\\\
&=\frac{\bar{z}}{z}\
B(\Delta_{1}-1,\Delta_{2}-1)\bigg{[}-P_{-1,-1}+z\big{(}c_{1}\hskip
1.13791ptJ^{0}_{-1}P_{-1,-1}+c_{2}P_{-2,-1}\big{)}\\\
&+z^{2}\bigg{(}c_{3}\hskip
1.13791ptJ^{0}_{-2}P_{-1,-1}+c_{4}P_{-3,-1}+c_{5}\big{(}2L_{-1}P_{-2,-1}-2\bar{L}_{-1}P_{-3,0}+2L_{-1}\bar{L}_{-1}P_{-2,0}-L^{2}_{-1}P_{-1,-1}\big{)}\bigg{)}\\\
&+\mathcal{O}(z^{3})\bigg{]}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)+\cdots\end{split}$
(13)

We now show that both sides of (13) transform in the same way under the action
of the Virasoro generator $L_{2}$. For this we take the gravitons in the OPE
to be Virasoro primaries so that they satisfy 333Here $(h,\bar{h})$ denote the
holomorphic and antiholomorphic conformal dimensions given by
$h=\frac{\Delta+\sigma}{2}$ and $\bar{h}=\frac{\Delta-\sigma}{2}$ where
$\sigma$ is the helicity of the graviton.

$\begin{split}&[L_{2},G_{h,\bar{h}}(z,\bar{z})]=z^{2}(z\partial_{z}+3h)G_{h,\bar{h}}(z,\bar{z})\end{split}$
(14)

Now we apply $L_{2}$ to the LHS of (13) and obtain using (14)

$\begin{split}&[L_{2},G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)]=z^{2}(z\partial_{z}+3h_{1})G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)\sim\mathcal{O}(\bar{z}z)+\cdots\end{split}$
(15)

where the dots denote terms of $\mathcal{O}(\bar{z}z^{2})$ and higher. The
leading $\mathcal{O}(\bar{z}z)$ term in (15) is given by,

$\begin{split}[L_{2},G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)]\Big{|}_{\mathcal{O}(z\bar{z})}&=z\bar{z}\
B(\Delta_{1}-1,\Delta_{2}-1)\
(1-3h_{1})P_{-1,-1}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)\\\ &=-z\bar{z}\
B(\Delta_{1}-1,\Delta_{2}-1)\
\frac{(3\Delta_{1}+4)}{2}P_{-1,-1}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)\end{split}$
(16)

Now we apply to $L_{2}$ to the RHS of the OPE (13). Using the following
commutation relations

$\begin{split}&[L_{m},J^{a}_{n}]=-nJ^{a}_{m+n},\quad[L_{m},P_{r,s}]=\left(\frac{m-1}{2}-r\right)P_{r+m,s},\\\
&[J^{a}_{m},J^{b}_{n}]=(a-b)J^{a+b}_{m+n},\quad[P_{m,n},P_{r,s}]=0\end{split}$
(17)

and the values of the coefficients $c_{3},c_{4},c_{5}$ given in (12) we find
that the action of $L_{2}$ on the R.H.S. of (13) yields

$\begin{split}&z\bar{z}\
B(\Delta_{1}-1,\Delta_{2}-1)\left[(\Delta_{1}+\Delta_{2}-2)c_{3}+\frac{7c_{4}}{2}-3c_{5}(\Delta_{1}+\Delta_{2}-2)\right]P_{-1,-1}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)+\cdots\\\
&=-z\bar{z}\ B(\Delta_{1}-1,\Delta_{2}-1)\
\frac{(3\Delta_{1}+4)}{2}P_{-1,-1}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)+\cdots\end{split}$
(18)

where the dots denote terms higher order in $z$. We can see that the
$\mathcal{O}(\bar{z}z)$ terms in (16) and (18) match precisely as expected
from the Virasoro invariance of the OPE (11).

In the appendix we perform more checks of the Virasoro invariance of the
celestial OPE of $++$ and $+-$ helicity gravitons in the MHV sector.

###  2.2 Reorganising the OPE in terms of Virasoro primaries and descendants

We have shown that up to $\mathcal{O}(\bar{z}z)$ terms the OPE of positive
helicity outgoing gravitons in the MHV sector given by (11) is invariant under
the (holomorphic) Virasoro symmetry. This implies that this OPE can be
reorganised according to representations of the Virasoro symmetry algebra as
is usually done in $2$-$d$ CFTs. In other words, we want to show that every
term at a particular order in the OPE is either a Virasoro primary or a
Virasoro descendant of another Virasoro primary which appears at a lower order
in $z$. The OPE coefficient multiplying a Virasoro primary cannot be obtained
by using the Virasoro symmetry alone but the OPE coefficient multiplying a
Virasoro descendant is uniquely determined by the Virasoro symmetry and the
OPE coefficient of the Virasoro primary from which it descends. For simplicity
here we reorganise only the $\mathcal{O}(\bar{z}z^{n})$ terms in the OPE.

Now in terms of Virasoro primaries and descendants, we find that the positive
helicity OPE (11) can be expressed as

$\begin{split}&G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)\Big{|}_{MHV}\\\
&=\frac{\bar{z}}{z}\
B(\Delta_{1}-1,\Delta_{2}-1)\bigg{[}C_{12\phi_{1}}\left(1+z\hskip
1.13791pt\beta_{\phi_{1}}^{(1)}L_{-1}+z^{2}\beta_{\phi_{1}}^{(1,1)}L_{-1}^{2}+z^{2}\beta_{\phi_{1}}^{(2)}{\color[rgb]{0,0,1}L_{-2}}\right)\phi_{1}(0)\\\
&+z\hskip 0.85355ptC_{12\phi_{2}}\left(1+z\hskip
1.13791pt\beta_{\phi_{2}}^{(1)}L_{-1}\right)\phi_{2}(0)+z\hskip
0.85355ptC_{12\phi_{3}}\left(1+z\hskip
1.13791pt\beta_{\phi_{3}}^{(1)}L_{-1}\right)\phi_{3}(0)+z^{2}\hskip
0.85355ptC_{12\phi_{4}}\phi_{4}(0)+\cdots\bigg{]}+\cdots\end{split}$ (19)

where $\phi_{i}(0)$ with $i=1,2,3,4$ are Virasoro primaries. Thus they satisfy
the conditions

$L_{0}\phi_{i}(0)=h_{\phi_{i}}\phi_{i}(0),\quad L_{m}\phi_{i}(0)=0,\quad m\geq
1$ (20)

where $h_{\phi_{i}}$ is the scaling dimension of the Virasoro primary
$\phi_{i}(0)$.

The coefficients $C_{12\phi_{i}}$ in (19) are given by

$\begin{split}&C_{12\phi_{1}}=-1,\quad
C_{12\phi_{2}}=\frac{\Delta_{1}-1}{\Delta_{1}+\Delta_{2}-2},\quad
C_{12\phi_{3}}=\frac{\Delta_{1}-\Delta_{2}}{\Delta_{1}+\Delta_{2}-2},\quad
C_{12\phi_{4}}=-\frac{\Delta_{1}\Delta_{2}}{\Delta_{1}+\Delta_{2}-1}\end{split}$
(21)

The $\beta_{\phi_{i}}^{(\vec{k})}$’s in (19) are completely determined by the
Virasoro algebra and are given by

$\begin{split}&\beta_{\phi_{i}}^{(1)}=\frac{h_{1}-h_{2}+h_{\phi_{i}}}{2h_{\phi_{i}}},\\\
&\beta_{\phi_{i}}^{(1,1)}=\frac{4h_{\phi_{i}}\left(2(h_{1}-h_{2}+h_{\phi_{i}})^{2}+h_{2}-4h_{1}-h_{\phi_{i}}\right)+c(h_{1}-h_{2}+h_{\phi_{i}})(h_{1}-h_{2}+h_{\phi_{i}}+1)}{4h_{\phi_{i}}\left(c+2ch_{\phi_{i}}+2h_{\phi_{i}}(8h_{\phi_{i}}-5)\right)}\\\
&\beta_{\phi_{i}}^{(2)}=\frac{h_{\phi_{i}}(2h_{1}+2h_{2}+h_{\phi_{i}}-1)+h_{1}+h_{2}-3(h_{1}-h_{2})^{2}}{\left(c+2ch_{\phi_{i}}+2h_{\phi_{i}}(8h_{\phi_{i}}-5)\right)}\end{split}$
(22)

where $h_{1},h_{2}$ are the holomorphic weights of the primary operators whose
OPE is being considered and $h_{\phi_{i}}$ is the holomorphic weight of the
Virasoro primary $\phi_{i}(0)$ which appears in this OPE. In our case

$\begin{split}&h_{1}=\frac{\Delta_{1}+2}{2},\quad
h_{2}=\frac{\Delta_{2}+2}{2}\end{split}$ (23)

Finally, the explicit expressions for the Virasoro primaries $\phi_{i}(0)$ are
as follows

$\begin{split}&\phi_{1}(0)=P_{-1,-1}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0),\quad\phi_{2}(0)=\bar{L}_{-1}P_{-2,0}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0),\\\
&\phi_{3}(0)=\left(P_{-2,-1}-\frac{2}{(\Delta_{1}+\Delta_{2}+2)}L_{-1}P_{-1,-1}\right)G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)\\\
&\phi_{4}(0)=\bigg{(}P_{-3,-1}+a_{1}J^{0}_{-2}P_{-1,-1}+a_{2}{\color[rgb]{0,0,1}L_{-2}}P_{-1,-1}+a_{3}L^{2}_{-1}P_{-1,-1}\\\
&\hskip
56.9055pt+a_{4}L_{-1}P_{-2,-1}+a_{5}L_{-1}\bar{L}_{-1}P_{-2,0}+a_{6}\bar{L}_{-1}P_{-3,0}\bigg{)}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)\end{split}$
(24)

where the coefficients denoted by $a_{i}$ and $b_{i}$ are given by

$\begin{split}&a_{1}=\frac{2(\Delta_{1}-1)(\Delta_{2}-1)}{C_{12\phi_{4}}(\Delta_{1}+\Delta_{2}-2)(\Delta_{1}+\Delta_{2}-1)},\quad
a_{2}=-\frac{C_{12\phi_{1}}\ \beta_{\phi_{1}}^{(2)}}{C_{12\phi_{4}}},\\\
&a_{3}=\frac{1}{C_{12\phi_{4}}}\left(-\frac{\Delta_{1}(\Delta_{1}-1)}{2(\Delta_{1}+\Delta_{2}-2)(\Delta_{1}+\Delta_{2}-1)}-C_{12\phi_{1}}\
\beta_{\phi_{1}}^{(1,1)}+\frac{2\hskip 0.85355ptC_{12\phi_{3}}\
\beta_{\phi_{3}}^{(1)}}{(\Delta_{1}+\Delta_{2}+2)}\right),\\\
&a_{4}=\frac{1}{C_{12\phi_{4}}}\left(\frac{\Delta_{1}(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}-2)(\Delta_{1}+\Delta_{2}-1)}-C_{12\phi_{3}}\
\beta_{\phi_{3}}^{(1)}\right),\\\
&a_{5}=\frac{1}{C_{12\phi_{4}}}\left(\frac{\Delta_{1}(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}-2)(\Delta_{1}+\Delta_{2}-1)}-C_{12\phi_{2}}\
\beta_{\phi_{2}}^{(1)}\right),\\\
&a_{6}=-\frac{\Delta_{1}(\Delta_{1}-1)}{C_{12\phi_{4}}(\Delta_{1}+\Delta_{2}-2)(\Delta_{1}+\Delta_{2}-1)}\end{split}$
(25)

As we have already mentioned we are not able to determine the value of the
Virasoro central charge $c$ in this way. The reason being that the
$\mathcal{O}(\bar{z}z)$ term in the OPE (11) does not contain the $L_{-2}$
descendant of the graviton primary and so both sides of the OPE (11) transform
in the same way under $L_{2}$ irrespective of the value of the central charge
$c$. This is also evident from the decomposition (19) which holds for
arbitrary value of the central charge $c$. In particular the states
$\phi_{i}(0)$ are Virasoro primaries for any value of the central charge as
can be easily checked. We leave the determination of the value of the central
charge to future research. 444Since we are considering tree level scattering
amplitudes, presumably the value of the central charge $c$ is zero.

##  3 Discussion

Since the MHV sector is manifestly Virasoro invariant, the dual theory which
computes the MHV scattering amplitudes has a stress tensor $T(z)$ whose modes
are the Virasoro generators $L_{n}$. In celestial CFT the stress tensor is
given by Kapec:2014opa ; Kapec:2016jld ; He:2017fsb  the shadow of the
subleading conformally soft graviton. In particular the holomorphic stress
tensor $T(z)$ is the shadow of the negative helicity subleading soft graviton.
Let us try to explain the appearance of $T(z)$ in the MHV sector along this
line.

So consider the NMHV amplitudes of the form $\langle{---++\cdots+}\rangle$.
Now if we make one of the negative helicity gravitons subleading (conformally)
soft Cachazo:2014fwa ; Donnay:2018neh  then the NMHV amplitude soft factorises
via the MHV amplitude $\langle{--++\cdots+}\rangle$ and this can explain the
appearance of the stress tensor $T(z)$ in the MHV sector. But for various
reasons this explanation cannot be complete. For example, the negative
helicity subleading soft graviton gives rise to $SL(2,\mathbb{C})$ current
algebra and its shadow gives the holomorphic stress tensor $T(z)$. Although we
find the stress tensor $T(z)$ in the MHV sector, the $SL(2,\mathbb{C})$
current algebra is not visible in the MHV sector. It is not clear that why
these two symmetry algebras arising from the negative helicity subleading soft
graviton are not treated on an equal footing. Similarly, we also have positive
helicity soft graviton in the MHV sector and we do find the
$\overline{SL(2,\mathbb{C})}$ current algebra symmetry Banerjee:2020zlg  in
the MHV sector but not the $\overline{\text{Virasoro}}$ symmetry. It may be
that not all symmetries are manifest Strominger:2021lvk  at the level of OPE.
We hope to return to these issues in future.

##  4 Acknowledgements

We are grateful to Andy Strominger for very helpful correspondence on related
topics. SB would also like to thank the string theory group in IISER Bhopal,
India for their hospitality during the final stages of the work. SG would like
to thank Yasha Neiman for useful discussions. The work of SB is partially
supported by the Science and Engineering Research Board (SERB) grant
MTR/2019/000937 (Soft-Theorems, S-matrix and Flat-Space Holography). The work
of SG is supported by the Quantum Gravity Unit of the Okinawa Institute of
Science and Technology Graduate University (OIST). The work of PP is partially
supported by a grant to CMI from the Infosys Foundation.

##  Appendix A Virasoro invariance of positive helicity OPE: further checks

In this section, we provide additional checks of the Virasoro invariance of
the positive helicity graviton OPE in the MHV sector. In particular we first
compute the $\mathcal{O}(z^{2})$ terms in the OPE in subsection A.1. The
invariance of the OPE at this order under the action of $L_{2}$ is proved in
subection A.2. In subsection A.3, we rewrite the terms of $\mathcal{O}(z^{n})$
upto $n=2$ in the positive helicity OPE according to of Virasoro primaries and
descendants.

###  A.1 $\mathcal{O}(z^{2})$ terms in positive helicity OPE

Here we compute the $\mathcal{O}(z^{2})$ terms in the OPE between positive
helicity graviton primaries in the MHV sector. The operators which can appear
at this order must have dimensions
$\left(\frac{7+h}{2},\frac{\bar{h}-1}{2}\right)$, where
$(h,\bar{h})=\left(\frac{\Delta_{1}+\Delta_{2}+1}{2},\frac{\Delta_{1}+\Delta_{2}-3}{2}\right)$.
We then assume the following ansatz for the $\mathcal{O}(z^{2})$ OPE555The
ansatz (26) is not the most general one, since it is possible to write down
additional descendants with dimensions
$\left(\frac{7+h}{2},\frac{\bar{h}-1}{2}\right)$. As we will see in this
section, the coefficients $x_{1},x_{2},x_{3},x_{4}$ in (26) can be uniquely
determined using the underlying symmetry algebras. This justifies why it is
sufficient to consider (26).

$\begin{split}&G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)\Big{|}_{\mathcal{O}(z^{2})}\\\
&=z^{2}B(\Delta_{1}-1,\Delta_{2}-1)\bigg{[}x_{1}\hskip
1.13791ptP_{-4,0}+x_{2}\hskip 1.13791ptJ^{1}_{-3}P_{-1,-1}+x_{3}\hskip
1.13791ptL_{-1}^{2}P_{-2,0}+x_{4}\hskip
1.13791ptL_{-1}P_{-3,0}\bigg{]}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)\end{split}$
(26)

where $x_{1},x_{2},x_{3},x_{4}$ are as of yet undetermined coefficients. Note
that using the null state relations

$\displaystyle\left[L_{-1}P_{-2,0}+J^{1}_{-2}P_{-1,-1}-(\Delta_{1}+\Delta_{2})P_{-3,0}\right]G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)=0$
(27)
$\displaystyle\left[L_{-1}P_{-1,-1}+2J^{0}_{-1}P_{-1,-1}-(\Delta_{1}+\Delta_{2})P_{-2,-1}-\bar{L}_{-1}P_{-2,0}\right]G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)=0$
(28)

we can express (26) purely in terms of $\overline{SL(2,\mathbb{C})}$ current
algebra and supertranslation descendants. But for the purpose of determining
the coefficients $x_{i}$, it turns out to be more convenient to work with the
form of the OPE given in (26).

We will now compute the $x_{i}$’s by demanding that both sides of the OPE
transform in the same way under the action of the underlying symmetries. For
this, let us first act with the $\overline{SL(2,\mathbb{C})}$ current algebra
generator $J^{0}_{1}$ on both sides of (26).

The action of $J^{0}_{1}$ on a conformal primary $\phi_{h,\bar{h}}(z,\bar{z})$
is given by

$\begin{split}&[J^{0}_{1},\phi_{h,\bar{h}}(z,\bar{z})]=z(\bar{z}\partial_{\bar{z}}+\bar{h})\phi_{h,\bar{h}}(z,\bar{z})\end{split}$
(29)

Using this we get

$\begin{split}&[J^{0}_{1},G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)]=z(\bar{z}\partial_{\bar{z}}+\bar{h}_{1})G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)\end{split}$
(30)

where $\bar{h}_{1}=(\Delta_{1}-2)/2$. Then plugging in the $\mathcal{O}(z)$
OPE into the R.H.S. of (30) yields

$\begin{split}&[J^{0}_{1},G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)]\Big{|}_{\mathcal{O}(z^{2})}\\\
&=z^{2}B(\Delta_{1}-1,\Delta_{2}-1)\hskip
1.13791pt\bar{h}_{1}\bigg{[}\Delta_{1}P_{-3,0}-\frac{(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}-2)}J^{1}_{-2}P_{-1,-1}\bigg{]}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)\end{split}$
(31)

Now applying $J^{0}_{1}$ to the R.H.S. of (26) and using the commutation
relations given in (17) we obtain

$\begin{split}&[J^{0}_{1},G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)]\Big{|}_{\mathcal{O}(z^{2})}\\\
&=z^{2}B(\Delta_{1}-1,\Delta_{2}-1)\bigg{[}\left(-\frac{x_{1}}{2}-2x_{4}+(\Delta_{1}+\Delta_{2})(\Delta_{1}+\Delta_{2}-4)x_{3}\right)P_{-3,0}\\\
&\hskip
28.45274pt+\left(-x_{2}-x_{3}(\Delta_{1}+\Delta_{2}-4)+\frac{x_{4}}{2}\right)J^{1}_{-2}P_{-1,-1}\bigg{]}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)\end{split}$
(32)

In arriving at (32), we have also used the null state relation (27). Now
invariance of the OPE under $J^{0}_{1}$ implies that (31) and (32) should
match. As a result we get

$\begin{split}&-\frac{x_{1}}{2}-2x_{4}+(\Delta_{1}+\Delta_{2})(\Delta_{1}+\Delta_{2}-4)x_{3}=\bar{h}_{1}\Delta_{1}\end{split}$
(33)

and

$\begin{split}&x_{2}+x_{3}(\Delta_{1}+\Delta_{2}-4)-\frac{x_{4}}{2}=\frac{\bar{h}_{1}(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}-2)}\end{split}$
(34)

We need a few more equations to solve for all the $x_{i}$’s. So, let us act
with the global spacetime translation generator $P_{0,-1}$ on both sides of
(26). $P_{0,-1}$ acts on a outgoing conformal primary
$\phi_{h,\bar{h}}(z,\bar{z})$ as follows

$\begin{split}&[P_{0,-1},\phi_{h,\bar{h}}(z,\bar{z})]=z\phi_{h+1/2,\bar{h}+1/2}(z,\bar{z})\end{split}$
(35)

(35) then implies

$\begin{split}&[P_{0,-1},G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)]=zG^{+}_{\Delta_{1}+1}(z,\bar{z})G^{+}_{\Delta_{2}}(0)\end{split}$
(36)

Inserting the $\mathcal{O}(z)$ OPE in the R.H.S. of (36) then gives

$\begin{split}&[P_{0,-1},G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)]\Big{|}_{\mathcal{O}(z^{2})}\\\
&=z^{2}B(\Delta_{1},\Delta_{2}-1)\bigg{[}(\Delta_{1}+1)P_{-3,0}-\frac{\Delta_{1}}{(\Delta_{1}+\Delta_{2}-1)}J^{1}_{-2}P_{-1,-1}\bigg{]}G^{+}_{\Delta_{1}+\Delta_{2}}(0)\end{split}$
(37)

Now evaluating the action of $P_{0,-1}$ on the R.H.S. of (26) we get

$\begin{split}&[P^{0}_{1},G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)]\Big{|}_{\mathcal{O}(z^{2})}\\\
&=z^{2}B(\Delta_{1}-1,\Delta_{2}-1)\bigg{[}(-x_{2}+x_{4}+2x_{3}(\Delta_{1}+\Delta_{2}+1))P_{-3,0}-2x_{3}\hskip
1.13791ptJ^{1}_{-2}P_{-1,-1}\bigg{]}G^{+}_{\Delta_{1}+\Delta_{2}}(0)\end{split}$
(38)

where we have once again used the null state (27). Now (37) and (38) should
match in order for the $\mathcal{O}(z^{2})$ OPE to be invariant under
$P_{0,-1}$. Therefore, we get

$\begin{split}&x_{4}+2x_{3}(\Delta_{1}+\Delta_{2}+1)-x_{2}=\frac{(\Delta_{1}+1)(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}-2)}\end{split}$
(39)

and

$\begin{split}&x_{3}=\frac{\Delta_{1}(\Delta_{1}-1)}{2(\Delta_{1}+\Delta_{2}-1)(\Delta_{1}+\Delta_{2}-2)}\end{split}$
(40)

Equations (34) and (39) can be simultaneously solved to give

$\begin{split}&x_{2}=-\frac{(\Delta_{1}-1)(\Delta_{2}-1)}{(\Delta_{1}+\Delta_{2}-1)(\Delta_{1}+\Delta_{2}-2)}\end{split}$
(41)

$\begin{split}&x_{4}=-\frac{\Delta_{1}(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}-1)(\Delta_{1}+\Delta_{2}-2)}\end{split}$
(42)

Plugging these results in (33) then yields

$\begin{split}&x_{1}=\frac{\Delta_{1}\Delta_{2}}{\Delta_{1}+\Delta_{2}-1}\end{split}$
(43)

The $\mathcal{O}(z^{2})$ OPE (26) then takes the form

$\begin{split}G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)\Big{|}_{\mathcal{O}(z^{2})}&=z^{2}B(\Delta_{1}-1,\Delta_{2}-1)\bigg{[}x_{1}\hskip
1.13791ptP_{-4,0}+x_{2}\hskip 1.13791ptJ^{1}_{-3}P_{-1,-1}\\\ &\hskip
42.67912pt+x_{3}\left(L_{-1}^{2}P_{-2,0}-2L_{-1}P_{-3,0}\right)\bigg{]}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)\end{split}$
(44)

where

$\begin{split}&x_{1}=\frac{\Delta_{1}\Delta_{2}}{\Delta_{1}+\Delta_{2}-1},\quad
x_{2}=-\frac{(\Delta_{1}-1)(\Delta_{2}-1)}{(\Delta_{1}+\Delta_{2}-1)(\Delta_{1}+\Delta_{2}-2)},\\\
&x_{3}=\frac{\Delta_{1}(\Delta_{1}-1)}{2(\Delta_{1}+\Delta_{2}-1)(\Delta_{1}+\Delta_{2}-2)}\end{split}$
(45)

###  A.2 $L_{2}$ invariance of $\mathcal{O}(z^{2})$ terms in positive helicity
OPE

We will now show that the $\mathcal{O}(z^{2})$ OPE computed in the previous
section and given by (44) is invariant under the action of $L_{2}$. In order
to do this we start with the Virasoro transformation property of graviton
primaries (14)

$\begin{split}&[L_{2},G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)]=z^{2}(z\partial_{z}+3h_{1})G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)\end{split}$
(46)

Substituting the $\mathcal{O}(z^{0}\bar{z}^{0})$ OPE into the R.H.S. of (46)
then yields

$\begin{split}[L_{2},G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)]\Big{|}_{\mathcal{O}(z^{2})}&=z^{2}B(\Delta_{1}-1,\Delta_{2}-1)\
3h_{1}P_{-2,0}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)\\\
&=z^{2}B(\Delta_{1}-1,\Delta_{2}-1)\
\frac{3(\Delta_{1}+2)}{2}P_{-2,0}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)\end{split}$
(47)

Now using the commutation relations (17) and the values of $x_{1},x_{2},x_{3}$
from (45) we find that the action of $L_{2}$ on the R.H.S. of (44) gives666In
order to derive (48) we have also used the relation
$\left(J^{1}_{-1}P_{-1,-1}-(\Delta-1)P_{-2,0}\right)G^{+}_{\Delta}=0$.

$\begin{split}&z^{2}B(\Delta_{1}-1,\Delta_{2}-1)\left[\frac{9x_{1}}{2}+3x_{2}(\Delta_{1}+\Delta_{2}-2)+3x_{3}(\Delta_{1}+\Delta_{2}-2)\right]P_{-2,0}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)\\\
&=z^{2}B(\Delta_{1}-1,\Delta_{2}-1)\
\frac{3(\Delta_{1}+2)}{2}P_{-2,0}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)\end{split}$
(48)

This matches with (47) and hence exhibits the Virasoro invariance of the
$\mathcal{O}(z^{2})$ terms in the OPE (44). Let us again note here that as in
the $\mathcal{O}(z\bar{z})$ OPE, the $L_{-2}$ descendant of
$G^{+}_{\Delta_{1}+\Delta_{2}}$ does not appear in the OPE (44). Therefore our
calculation here is insensitive to the central charge of the Virasoro algebra
and Virasoro invariance of (44) continues to hold for any value of the central
charge.

###  A.3 Reorganising OPE in terms of Virasoro primaries and descendants

In section 2.2 we showed that the $\mathcal{O}(\bar{z}z^{n})$ terms in the
positive helicity graviton OPE in the MHV sector can be reorganised according
to Virasoro primaries and descendants. Here we provide another example of this
reorganisation by considering terms of $\mathcal{O}(z^{n})$ in the OPE. So,
let us begin with the following set of terms in the OPE,

$\begin{split}&G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)\Big{|}_{MHV}\\\
&=B(\Delta_{1}-1,\Delta_{2}-1)\bigg{[}P_{-2,0}+z\big{(}d_{1}P_{-3,0}+d_{2}J^{1}_{-2}P_{-1,-1}\big{)}\\\
&+z^{2}\big{(}x_{1}P_{-4,0}+x_{2}J^{1}_{-3}P_{-1,-1}+x_{3}\left(L_{-1}^{2}P_{-2,0}-2L_{-1}P_{-3,0}\right)\big{)}+\mathcal{O}\left(z^{3}\right)\bigg{]}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)+\cdots\end{split}$
(49)

where $d_{1},d_{2}$ in (50) were obtained using symmetries in Banerjee:2020zlg
and are given by

$\begin{split}&d_{1}=\Delta_{1},\quad
d_{2}=-\frac{\Delta_{1}-1}{\Delta_{1}+\Delta_{2}-2}\end{split}$ (50)

$x_{1},x_{2},x_{3}$ are given in (45). Now (49) can also be written as

$\begin{split}&G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0)\Big{|}_{MHV}\\\
&=B(\Delta_{1}-1,\Delta_{2}-1)\bigg{[}C_{12\psi_{1}}\left(1+z\hskip
1.13791pt\beta_{\psi_{1}}^{(1)}L_{-1}+z^{2}\beta_{1}^{(1,1)}L_{-1}^{2}+z^{2}\beta_{\psi_{1}}^{(2)}L_{-2}\right)\psi_{1}(0)\\\
&+z\hskip 0.85355ptC_{12\psi_{2}}\left(1+z\hskip
1.13791pt\beta_{\psi_{2}}^{(1)}L_{-1}\right)\psi_{2}(0)+z^{2}\hskip
0.85355ptC_{12\psi_{3}}\psi_{3}(0)+\mathcal{O}\left(z^{3}\right)\bigg{]}+\cdots\end{split}$
(51)

where $\psi_{i}(0)$ with $i=1,2,3$ are Virasoro primaries for arbitrary values
of the central charge. Their explicit expressions are

$\begin{split}&\psi_{1}(0)=P_{-2,0}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0),\quad\psi_{2}(0)=\left(P_{-3,0}-\frac{3}{(\Delta_{1}+\Delta_{2}+4)}L_{-1}P_{-2,0}\right)G^{+}_{\Delta_{1}+\Delta_{2}-1}(0),\\\
&\psi_{3}(0)=\bigg{(}P_{-4,0}+b_{1}J^{1}_{-3}P_{-1,-1}+b_{2}L_{-1}P_{-3,0}+b_{3}L_{-1}^{2}P_{-2,0}+b_{4}L_{-2}P_{-2,0}\bigg{)}G^{+}_{\Delta_{1}+\Delta_{2}-1}(0)\end{split}$
(52)

where the $b_{i}$’s in (52) are given by

$\begin{split}&b_{1}=-\frac{(\Delta_{1}-1)(\Delta_{2}-1)}{C_{12\psi_{3}}(\Delta_{1}+\Delta_{2}-1)(\Delta_{1}+\Delta_{2}-2)}\\\
&b_{2}=\frac{1}{C_{12\psi_{3}}}\bigg{(}-\frac{(\Delta_{1}-1)(\Delta_{2}-1)}{(\Delta_{1}+\Delta_{2}-1)(\Delta_{1}+\Delta_{2}-2)}-C_{12\psi_{2}}\
\beta_{\psi_{2}}^{(1)}\bigg{)}\\\
&b_{3}=\frac{1}{C_{12\psi_{3}}}\bigg{(}\frac{(\Delta_{1}-1)(\Delta_{2}-1)}{2(\Delta_{1}+\Delta_{2}-1)(\Delta_{1}+\Delta_{2}-2)}-C_{12\psi_{1}}\
\beta_{\psi_{1}}^{(1,1)}+\frac{3\hskip 0.85355ptC_{12\psi_{2}}\
\beta_{\psi_{2}}^{(1)}}{(\Delta_{1}+\Delta_{2}+4)}\bigg{)}\\\
&b_{4}=-\frac{C_{12\psi_{1}}\
\beta_{\psi_{1}}^{(2)}}{C_{12\psi_{3}}}\end{split}$ (53)

The coefficients $C_{12\psi_{i}}$ in (51) are given by

$\begin{split}&C_{12\psi_{1}}=1,\quad
C_{12\psi_{2}}=\frac{\Delta_{2}-\Delta_{1}}{\Delta_{1}+\Delta_{2}-2},\quad
C_{12\psi_{3}}=\frac{\Delta_{1}\Delta_{2}}{\Delta_{1}+\Delta_{2}-1}\end{split}$
(54)

Note that the $C_{12\psi_{i}}$’s cannot be determined by using only the
Virasoro algebra. The coefficients $\beta^{(\vec{k})}_{\psi_{i}}$ in (51) are
uniquely fixed by the Virasoro algebra. The explicit formulae for the
$\beta^{(\vec{k})}_{\psi_{i}}$’s take the same form as in (22) with
$h_{\phi_{i}}$ replaced by $h_{\psi_{i}}$, where $h_{\psi_{i}}$ denotes the
holomorphic weight of the Virasoro primary $\psi_{i}$ appearing in the R.H.S.
of (51).

##  Appendix B Virasoro invariance of mixed helicity OPE

In this section of the Appendix, we consider the celestial OPE between an
outgoing positive helicity graviton primary and an outgoing negative helicity
graviton primary in the MHV sector. Here we show that this mixed helicity OPE
is also Virasoro invariant. In order to furnish a non-trivial check of
Virasoro invariance we first compute the $\mathcal{O}(z\bar{z})$ terms in the
mixed helicity OPE in subsection B.1. This is followed by an explicit check of
the invariance of this OPE under the action of $L_{2}$ in subsection B.2. In
subsection B.3 it is shown that the mixed helicity OPE can be organised
according to Virasoro representations.

###  B.1 $\mathcal{O}(z\bar{z})$ term in mixed helicity OPE using symmetries

The operators which can appear at $\mathcal{O}(z\bar{z})$ in the mixed
helicity OPE should have weights
$\left(\frac{3+h}{2},\frac{\bar{h}+1}{2}\right)$, where
$(h,\bar{h})=\left(\frac{\Delta_{1}+\Delta_{2}-3}{2},\frac{\Delta_{1}+\Delta_{2}+1}{2}\right)$.
Let us then consider the OPE at this order to take the following form

$\begin{split}&G^{+}_{\Delta_{1}}(z,\bar{z})G^{-}_{\Delta_{2}}(0)\Big{|}_{\mathcal{O}(z\bar{z})}\\\
&=z\bar{z}\
B(\Delta_{1}-1,\Delta_{2}+3)\bigg{[}\bigg{(}\alpha_{1}J^{0}_{-2}P_{-1,-1}+\alpha_{2}P_{-3,-1}+\alpha_{3}J^{0}_{-1}P_{-2,-1}+\alpha_{4}(J^{0}_{-1})^{2}P_{-1,-1}\\\
&+\alpha_{5}\bar{L}_{-1}P_{-3,0}+\alpha_{6}\bar{L}_{-1}J^{0}_{-1}P_{-2,0}+\alpha_{7}J^{-1}_{-1}P_{-2,0}+\alpha_{8}\bar{L}_{-1}J^{1}_{-2}P_{-1,-1}\bigg{)}G^{-}_{\Delta_{1}+\Delta_{2}-1}(0)\\\
&+\bigg{(}\alpha_{9}P_{-2,-1}^{2}+\alpha_{10}\bar{L}_{-1}P_{-2,-1}P_{-2,0}+\alpha_{11}\bar{L}_{-1}^{2}P_{-2,0}^{2}\bigg{)}G^{-}_{\Delta_{1}+\Delta_{2}-2}(0)\bigg{]}\end{split}$
(55)

where $\alpha_{i},i=1,2,\ldots,11$ are as of yet unknown coefficients. In
order to determine these using symmetries we follow the same procedure as in
Section A.1. We impose that both sides of the OPE (55) transform identically
under the action of supertranslations and the $\overline{SL(2,\mathbb{C})}$
current algebra. This yields a system of equations for the $\alpha_{i}$’s
which can then be simultaneously solved. In particular we will apply below the
generators $P_{0,-1},P_{-1,0}$ and $J^{1}_{0}$ for deriving these equations.

First let us consider acting with $P_{0,-1}$. Demanding both sides of (55) to
transform in the same fashion under the action of $P_{0,-1}$ leads to

$\displaystyle\alpha_{4}=-\frac{2\Delta_{1}(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}+2)(\Delta_{1}+\Delta_{2}+3)}$
(56)
$\displaystyle-\frac{\alpha_{1}}{2}-\frac{\alpha_{3}}{2}+\frac{\alpha_{4}}{4}=-\frac{(\Delta_{1}+1)(\Delta_{1}-1)}{\Delta_{1}+\Delta_{2}+2}$
(57) $\displaystyle\alpha_{6}+2\alpha_{8}=0$ (58)

Next we act with $P_{-1,0}$ on both sides of the OPE (55). In this case we
obtain

$\displaystyle\alpha_{6}=-\alpha_{4}=\frac{2\Delta_{1}(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}+2)(\Delta_{1}+\Delta_{2}+3)}$
(59)
$\displaystyle\alpha_{8}=-\frac{\Delta_{1}(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}+2)(\Delta_{1}+\Delta_{2}+3)}$
(60)
$\displaystyle\alpha_{11}=-\frac{\alpha_{6}}{4}=-\frac{\Delta_{1}(\Delta_{1}-1)}{2(\Delta_{1}+\Delta_{2}+2)(\Delta_{1}+\Delta_{2}+3)}$
(61)
$\displaystyle\frac{\alpha_{1}}{2}+\frac{\alpha_{4}}{4}+\alpha_{5}-\alpha_{8}=\frac{(\Delta_{1}+1)(\Delta_{1}-1)}{\Delta_{1}+\Delta_{2}+2}$
(62)
$\displaystyle\frac{\alpha_{3}}{2}-\frac{\alpha_{6}}{2}+\alpha_{7}+\alpha_{10}=0$
(63)

Finally invariance of the OPE under the action of $J^{1}_{0}$ yields

$\displaystyle\alpha_{1}+\alpha_{4}+(2\bar{h}-1)\alpha_{8}=-\frac{2\bar{h}_{1}(\Delta_{1}-1)}{\Delta_{1}+\Delta_{2}+2}$
(64)
$\displaystyle\alpha_{2}+\alpha_{3}+(2\bar{h}-1)\alpha_{5}=2\bar{h}_{1}\Delta_{1}$
(65)
$\displaystyle\alpha_{3}+2(\Delta+3)\alpha_{4}+(2\bar{h}-1)\alpha_{6}+2\alpha_{7}=0$
(66) $\displaystyle(\Delta+2)\alpha_{3}+2\alpha_{9}+(2\bar{h}-1)\alpha_{10}=0$
(67) $\displaystyle(\Delta+2)\alpha_{6}+\alpha_{10}+4(\bar{h}-1)\alpha_{11}=0$
(68)

The above system of equations can be solved to uniquely determine all the
$\alpha_{i}$’s. These are as follows

$\begin{split}&\alpha_{1}=\frac{(\Delta_{1}-1)(\Delta_{1}+2\Delta_{2}+6)}{(\Delta_{1}+\Delta_{2}+2)(\Delta_{1}+\Delta_{2}+3)},\quad\alpha_{2}=-\Delta_{1},\quad\alpha_{3}=\frac{2\Delta_{1}(\Delta_{1}-1)}{\Delta_{1}+\Delta_{2}+2},\\\
&\alpha_{4}=-\frac{2\Delta_{1}(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}+2)(\Delta_{1}+\Delta_{2}+3)},\quad\alpha_{5}=\frac{\Delta_{1}(\Delta_{1}-1)}{\Delta_{1}+\Delta_{2}+2},\quad\alpha_{6}=\frac{2\Delta_{1}(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}+2)(\Delta_{1}+\Delta_{2}+3)}\\\
&\alpha_{7}=\frac{\Delta_{1}(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}+2)(\Delta_{1}+\Delta_{2}+3)},\quad\alpha_{8}=-\frac{\Delta_{1}(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}+2)(\Delta_{1}+\Delta_{2}+3)}\\\
&\alpha_{9}=-\frac{\Delta_{1}(\Delta_{1}-1)}{2},\quad\alpha_{10}=-\frac{\Delta_{1}(\Delta_{1}-1)}{\Delta_{1}+\Delta_{2}+2},\quad\alpha_{11}=-\frac{\Delta_{1}(\Delta_{1}-1)}{2(\Delta_{1}+\Delta_{2}+2)(\Delta_{1}+\Delta_{2}+3)}\end{split}$
(69)

###  B.2 $L_{2}$ invariance of $\mathcal{O}(z\bar{z})$ terms in mixed helicity
OPE

We shall now verify that the mixed helicity OPE at $\mathcal{O}(z\bar{z})$
(55) is invariant under $L_{2}$. For this we act with $L_{2}$ on both sides of
(55). From the L.H.S. we get

$\begin{split}[L_{2},G^{+}_{\Delta_{1}}(z,\bar{z})G^{-}_{\Delta_{2}}(0)]\Big{|}_{\mathcal{O}(z\bar{z})}&=-z\bar{z}\
B(\Delta_{1}-1,\Delta_{2}+3)\
\frac{(3\Delta_{1}+4)}{2}P_{-1,-1}G^{-}_{\Delta_{1}+\Delta_{2}-1}(0)\end{split}$
(70)

The action of $L_{2}$ on the R.H.S. of (55) evaluates to

$\begin{split}&z\bar{z}\
B(\Delta_{1}-1,\Delta_{2}+3)\left[(\Delta_{1}+\Delta_{2}+2)\alpha_{1}+\frac{7\alpha_{2}}{2}+\alpha_{5}-\frac{\alpha_{6}}{2}\right]P_{-1,-1}G^{-}_{\Delta_{1}+\Delta_{2}-1}(0)\end{split}$
(71)

Using the values of $\alpha_{1},\alpha_{2},\alpha_{5},\alpha_{6}$ from (69),
we find that (71) simplifies to

$\begin{split}&-z\bar{z}\ B(\Delta_{1}-1,\Delta_{2}+3)\
\frac{(3\Delta_{1}+4)}{2}P_{-1,-1}G^{-}_{\Delta_{1}+\Delta_{2}-1}(0)\end{split}$
(72)

This matches with (70) and thereby proves that the $\mathcal{O}(z\bar{z})$
term in the mixed helicity OPE is Virasoro invariant. As in the case of the
positive helicity OPE, the central charge of the Virasoro algebra does not
play any role here.

###  B.3 Reorganising OPE in terms of Virasoro primaries and descendants

In the MHV sector, the mixed helicity graviton OPE takes the following form

$\begin{split}&G^{+}_{\Delta_{1}}(z,\bar{z})G^{-}_{\Delta_{2}}(0)\Big{|}_{MHV}\\\
&=\frac{\bar{z}}{z}\
B(\Delta_{1}-1,\Delta_{2}+3)\bigg{[}-P_{-1,-1}G^{-}_{\Delta_{1}+\Delta_{2}-1}(0)+z\big{(}\rho_{1}\hskip
1.13791ptJ^{0}_{-1}P_{-1,-1}+\rho_{2}P_{-2,-1}\big{)}G^{-}_{\Delta_{1}+\Delta_{2}-1}(0)\\\
&\hskip
28.45274pt+z^{2}\bigg{(}\alpha_{1}J^{0}_{-2}P_{-1,-1}+\alpha_{2}P_{-3,-1}+\alpha_{5}\left(2J^{0}_{-1}P_{-2,-1}+\bar{L}_{-1}P_{-3,0}\right)\\\
&\hskip
28.45274pt+\frac{\alpha_{6}}{2}\big{(}-2(J^{0}_{-1})^{2}P_{-1,-1}+2\bar{L}_{-1}J^{0}_{-1}P_{-2,0}+J^{-1}_{-1}P_{-2,0}-\bar{L}_{-1}J^{1}_{-2}P_{-1,-1}\big{)}\bigg{)}G^{-}_{\Delta_{1}+\Delta_{2}-1}(0)\\\
&\hskip
28.45274pt+z^{2}\big{(}\alpha_{9}P_{-2,-1}^{2}+\alpha_{10}\bar{L}_{-1}P_{-2,-1}P_{-2,0}+\alpha_{11}\bar{L}_{-1}^{2}P_{-2,0}^{2}\big{)}G^{-}_{\Delta_{1}+\Delta_{2}-2}(0)+\mathcal{O}\left(z^{3}\right)\bigg{]}\\\
&+B(\Delta_{1}-1,\Delta_{2}+3)\bigg{[}P_{-2,0}G^{-}_{\Delta}(0)+z\bigg{(}\eta_{1}P_{-3,0}+\eta_{2}J^{1}_{-2}P_{-1,-1}\bigg{)}G^{-}_{\Delta_{1}+\Delta_{2}-1}(0)+\mathcal{O}\left(z^{2}\right)\bigg{]}+\cdots\end{split}$
(73)

where $\rho_{1},\rho_{2}$ and $\eta_{1},\eta_{2}$ are given by

$\begin{split}&\rho_{1}=\frac{2(\Delta_{1}-1)}{\Delta_{1}+\Delta_{2}+2},\quad\rho_{2}=-\Delta_{1},\quad\eta_{1}=\Delta_{1},\quad\eta_{2}=-\frac{\Delta_{1}-1}{\Delta_{1}+\Delta_{2}+2}\end{split}$
(74)

The values of $\rho_{1},\rho_{2}$ were computed previously in Banerjee:2020zlg
. Although we have not shown in this paper, $\eta_{1},\eta_{2}$ can also be
easily obtained from symmetries. The $\alpha_{i}$’s are given by (69). The
dots in (73) denote further terms which we have suppressed for simplicity.

Now it is again possible to rearrange (73) in terms of Virasoro primaries and
descendants, so as to make the Virasoro invariance of the OPE manifest.
Carrying out this reorganisation we find the following result

$\begin{split}&G^{+}_{\Delta_{1}}(z,\bar{z})G^{-}_{\Delta_{2}}(0)\Big{|}_{MHV}\\\
&=\frac{\bar{z}}{z}\
B(\Delta_{1}-1,\Delta_{2}+3)\bigg{[}C_{12\varphi_{1}}\left(1+z\hskip
1.13791pt\beta_{\varphi_{1}}^{(1)}L_{-1}+z^{2}\beta_{\varphi_{1}}^{(1,1)}L_{-1}^{2}+z^{2}\beta_{\varphi_{1}}^{(2)}L_{-2}\right)\varphi_{1}(0)\\\
&+z\hskip 0.85355ptC_{12\varphi_{2}}\left(1+z\hskip
1.13791pt\beta_{\varphi_{2}}^{(1)}L_{-1}\right)\varphi_{2}(0)+z\hskip
0.85355ptC_{12\varphi_{3}}\left(1+z\hskip
1.13791pt\beta_{\varphi_{3}}^{(1)}L_{-1}\right)\varphi_{3}(0)+z^{2}\hskip
0.85355ptC_{12\varphi_{4}}\varphi_{4}(0)+\mathcal{O}\left(z^{3}\right)\bigg{]}\\\
&+B(\Delta_{1}-1,\Delta_{2}+3)\bigg{[}C_{12\Omega_{1}}\left(1+z\hskip
1.13791pt\beta_{\Omega_{1}}^{(1)}L_{-1}\right)\Omega_{1}(0)+z\hskip
0.85355ptC_{12\Omega_{2}}\hskip
1.13791pt\Omega_{2}(0)+\mathcal{O}\left(z^{2}\right)\bigg{]}+\cdots\end{split}$
(75)

where the operators denoted by $\varphi_{i}(0)$ and $\Omega_{i}(0)$ are
Virasoro primaries (for any value of the central charge). Their explicit
expressions are given by (77) and (78). The coefficients $C_{12\varphi_{i}}$
and $C_{12\Omega{i}}$ are

$\begin{split}&C_{12\varphi_{1}}=-1,\quad C_{12\varphi_{2}}=-\Delta_{1},\quad
C_{12\varphi_{3}}=-\Delta_{1},\quad C_{12\Omega_{1}}=1,\quad
C_{12\Omega_{2}}=\Delta_{1}\end{split}$ (76)

Once again it is worth noting that Virasoro symmetry alone does not fix the
values given in (76). The coefficients $\beta^{(\vec{k})}_{\varphi_{i}}$’s and
$\beta^{(\vec{k})}_{\Omega_{i}}$’s in (75), which are uniquely determined by
the Virasoro algebra, are given by similar expressions as in (22). Let us
finally note the expressions for the Virasoro primaries
$\varphi_{i}(0),\Omega_{i}(0)$.

$\begin{split}&\varphi_{1}(0)=P_{-1,-1}G^{-}_{\Delta_{1}+\Delta_{2}-1}(0),\\\
&\varphi_{2}(0)=\left(P_{-2,-1}-\frac{2(\Delta_{1}-1)}{\Delta_{1}(\Delta_{1}+\Delta_{2}+2)}J^{0}_{-1}P_{-1,-1}-\frac{(\Delta_{1}+1)}{\Delta_{1}(\Delta_{1}+\Delta_{2}-2)}L_{-1}P_{-1,-1}\right)G^{-}_{\Delta_{1}+\Delta_{2}-1}(0)\\\
&\varphi_{3}(0)=\bigg{[}P_{-3,-1}-\frac{(\Delta_{1}-1)(\Delta_{1}+2\Delta_{2}+6)}{\Delta_{1}(\Delta_{1}+\Delta_{2}+3)}J^{0}_{-2}P_{-1,-1}\\\
&\hskip
42.67912pt-\frac{(\Delta_{1}-1)}{\Delta_{1}+\Delta_{2}+2}\left(2J^{0}_{-1}P_{-2,-1}+\bar{L}_{-1}P_{-3,0}\right)\\\
&\hskip
42.67912pt+\frac{(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2}+2)(\Delta_{1}+\Delta_{2}+3)}\bigg{(}2(J^{0}_{-1})^{2}P_{-1,-1}-2\bar{L}_{-1}J^{0}_{-1}P_{-2,0}\\\
&\hskip
213.39566pt-J^{-1}_{-1}P_{-2,0}+\bar{L}_{-1}J^{1}_{-2}P_{-1,-1}\bigg{)}\bigg{]}G^{-}_{\Delta_{1}+\Delta_{2}-1}(0)\\\
&\hskip
42.67912pt+\bigg{(}\frac{(\Delta_{1}-1)}{2}P_{-2,-1}^{2}+\frac{(\Delta_{1}-1)}{\Delta_{1}+\Delta_{2}+2}\
\bar{L}_{-1}P_{-2,-1}P_{-2,0}\\\ &\hskip
71.13188pt+\frac{(\Delta_{1}-1)}{2(\Delta_{1}+\Delta_{2}+2)(\Delta_{1}+\Delta_{2}+3)}\bar{L}_{-1}^{2}P_{-2,0}^{2}\bigg{)}G^{-}_{\Delta_{1}+\Delta_{2}-2}(0)\\\
&\hskip
28.45274pt-\frac{(\Delta_{1}-4)(4\Delta_{1}+3)}{2\Delta_{1}(\Delta_{1}+\Delta_{2}-2)(4\Delta_{1}+\Delta_{2}-13)}L_{-1}^{2}P_{-1,-1}G^{-}_{\Delta_{1}+\Delta_{2}-1}(0)\\\
&\hskip
28.45274pt-\frac{(\Delta_{1}(3\Delta_{2}-8)+4\Delta_{2}-10)}{\Delta_{1}(\Delta_{1}+\Delta_{2}-2)(4\Delta_{1}+\Delta_{2}-13)}L_{-2}P_{-1,-1}G^{-}_{\Delta_{1}+\Delta_{2}-1}(0)-\frac{(\Delta_{1}+2)}{\Delta_{1}+\Delta_{2}}L_{-1}\varphi_{2}(0)\end{split}$
(77)

and

$\begin{split}&\Omega_{1}(0)=P_{-2,0}G^{-}_{\Delta_{1}+\Delta_{2}-1}(0),\\\
&\Omega_{2}(0)=\left(P_{-3,0}-\frac{(\Delta_{1}+2)}{\Delta_{1}(\Delta_{1}+\Delta_{2})}L_{-1}P_{-2,0}-\frac{(\Delta_{1}-1)}{\Delta_{1}(\Delta_{1}+\Delta_{2}+2)}J^{1}_{-2}P_{-1,-1}\right)G^{-}_{\Delta_{1}+\Delta_{2}-1}(0)\end{split}$
(78)

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