}T_{e}(t,\hat{z},\hat{\bar{z}})$ (109)

where the counter-clockwise contour $C_{u}^{\prime}$ encloses the entire upper
half of a $y=ax+b$ plane as well as the line $t=0$; thus it encloses the
projections of the singularities of the vector field $\hat{z}^{m+1}$.

Because the singularities of the vector fields $\hat{z}^{m+1}$ are independent
of $t$, this definition of $L_{m}$ under the $j\epsilon$-prescription is
practically equivalent to the definition (97) under the $\theta$-prescription
where the contour $C_{e}$ (on $x$–$y$ plane) must enclose the entire complex
$z$-plane (hence, the singularities of the vector field). As is evident from
the construction, both of these definitions should be thought to be valid only
inside correlators.

Thus the $j\epsilon$-prescription plays a very crucial rule to fix the
definitions of the (anti-)holomorphic quantum conserved charge operators but
not those of the mixed super-translation generators. As a very significant
byproduct, we get to establish the relation between the OPEs and the operator
commutation relations, e.g. (108), via complex contour integrals (on $S^{2}$).

The initial conditions like (45) have clearer technical meaning in the
$j\epsilon$-prescription. E.g. if we put $t\rightarrow-\infty$ in (107), all
the poles are pushed into the lower half of a $y=ax+b$ plane; these poles are
not then enclosed by the contours $C_{u}$ or $C_{u}^{\prime}$, hence
contributing nothing to the integral.

It may seem that all the above conclusions remain intact even if we replace
the $j\epsilon(t-t_{p})$ part of $\Delta\tilde{z}_{p}$ by
$j\lambda{\left(t-t_{p}\right)}^{2n+1}$ with
$\lambda=\lambda_{1}+j\lambda_{2}$ such that $\lambda_{1}>0$. But we wanted
the dimension of $\lambda$ to be that of speed, so we chose $n=0$. Moreover,
remembering that $\partial_{t}\langle
T_{e}(t,\hat{z},\hat{\bar{z}})X\rangle=0$ in the OPE limit, we are forced to
take $\lambda_{1}\rightarrow 0^{+}$ and $\lambda_{2}=0$ that reduces to the
$j\epsilon$-prescription.

We shall always write the OPEs after analytically continuing the expressions
involving $\hat{z},\hat{\bar{z}}$ back to ‘real’ (wrt complex unit $j$)
$z,\bar{z}$.

##  5 Symmetry Algebra from the OPE

We now proceed to find the symmetry algebra that is manifest at the level of
the $1+2$D Carrollian conformal OPEs. From the discussion in section 3, it
already appears that all of the $S^{\pm}_{0}$, $S^{\pm}_{1}$, $T(\bar{T})$
fields can not be simultaneously taken to form consistent mutual OPEs. In this
section, we pinpoint the reasons and choose an appropriate subset from these
six generator-fields that allows the formation of consistent OPEs. Under two
crucial assumptions (one is inspired from the usual 2D CFT
Zamolodchikov:1989mz  and the another from the Celestial CFT
Fotopoulos:2019tpe ), we are then able to fix the pole singularities of these
mutual OPEs using only the OPE-commutativity property, just as in
Zamolodchikov:1985wn ; Zamolodchikov:1989mz ; Saha:2022gjw ; Bagchi:2023dzx .
The ansatz for these OPEs are made from the corresponding Carrollian conformal
Ward identities (themselves derived from general symmetry principles) in the
first place. Our results for the singular parts of these OPEs match with those
obtained in the Celestial CFT Guevara:2019ypd ; Fotopoulos:2019vac ;
Guevara:2021abz ; Banerjee:2022wht ; Puhm:2019zbl  where the starting point
was the (linearized) Einstein theory in the bulk AFS. Finally, we translate
these OPEs into the language of the algebra of the modes, using the complex
contour integral prescription developed in section 4.

###  5.1 The OPEs

We begin by writing down the OPEs directly in the $j\epsilon$ form from the
corresponding Ward identities derived in section 3. The Bosonic or Fermionic
exchange properties of the two composite (or local) operators involved can be
readily implemented in this form. On the contrary, in the form involving the
temporal step-function, it is not straight-forward to use these exchange
properties in the OPE because the discontinuous initial conditions like (45)
needs to be simultaneously altered in the process.

As discussed in Saha:2022gjw , in a Carrollian field theory, two fields inside
a correlator are to be treated as a single composite operator if they are
inserted at the same spatial location (but possibly at different times)
causing the Carrollian-invariant norm to vanish151515The flat Carrollian-
invariant norm is the Euclidean (spatial) distance:
$\big{|}\vec{x}_{1}-\vec{x}_{2}\big{|}$ .. This is the essence of the spatial
absoluteness in Carrollian field theory. So, the two local fields whose
operator-product is considered are always inserted at different spatial
locations (thus, contact terms can not appear as OPE coefficients). Moreover,
to avoid the time-ordering ambiguity, they are also inserted at different
times.

We now note down various OPEs involving a primary Carrollian conformal
multiplet $\Phi(t_{p},\vec{x}_{p})$, with
$\Delta\tilde{z}_{p}:=z-z_{p}-j\epsilon(t-t_{p})$, :

$\displaystyle
P(t,z,\bar{z})\Phi(t_{p},z_{p},\bar{z}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left(\frac{\partial_{t_{p}}}{(\Delta\tilde{z}_{p})}+\frac{\bm{\xi}_{p}}{{(\Delta\tilde{z}_{p})}^{2}}\right)\Phi(t_{p},z_{p},\bar{z}_{p})$
(110) $\displaystyle
S^{+}_{0}(t,z,\bar{z})\Phi(t_{p},z_{p},\bar{z}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left\\{\frac{\bar{z}-\bar{z}_{p}}{(\Delta\tilde{z}_{p})}\partial_{t_{p}}+\frac{\bar{z}-\bar{z}_{p}}{{(\Delta\tilde{z}_{p})}^{2}}\bm{\xi}_{p}-\frac{\bar{{\bm{\xi}}}_{p}}{(\Delta\tilde{z}_{p})}\right\\}\Phi(t_{p},z_{p},\bar{z}_{p})$
(111) $\displaystyle
T(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{h_{p}}{(\Delta\tilde{z}_{p})^{2}}+\frac{\partial_{z_{p}}}{\Delta\tilde{z}_{p}}-\frac{t-t_{p}}{2}\left\\{\frac{\partial_{t_{p}}}{(\Delta\tilde{z}_{p})^{2}}+\frac{2\bm{\xi}_{p}}{{(\Delta\tilde{z}_{p})}^{3}}\right\\}\right]\Phi(\mathbf{x}_{p})$
(112) $\displaystyle
S_{1}^{+}(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-\frac{i}{2}\left[\frac{(\bar{z}-\bar{z}_{p})^{2}}{\Delta\tilde{z}_{p}}\partial_{\bar{z}_{p}}-2\bar{h}_{p}\frac{\bar{z}-\bar{z}_{p}}{\Delta\tilde{z}_{p}}+\frac{0}{\Delta\tilde{z}_{p}}\right.$
$\displaystyle\hskip
122.34685pt\left.+(t-t_{p})\left(\frac{\bar{z}-\bar{z}_{p}}{(\Delta\tilde{z}_{p})}\partial_{t_{p}}+\frac{\bar{z}-\bar{z}_{p}}{{(\Delta\tilde{z}_{p})}^{2}}\bm{\xi}_{p}-\frac{\bar{{\bm{\xi}}}_{p}}{(\Delta\tilde{z}_{p})}\right)\right]\Phi(\mathbf{x}_{p})$
(113)

where $\sim$ denotes ‘modulo terms holomorphic (regular) in
$\Delta\tilde{z}_{p}$ ’. That there are regular terms in these OPEs can be
easily checked by expanding the corresponding Ward identities in
$j\epsilon$-prescription as power series simultaneously in e.g.
${z}-{z}_{1}-j\epsilon(t-t_{1})$ and $\bar{z}-\bar{z}_{1}$ remembering that
$\big{|}z-z_{1}\big{|}<\min\limits_{p\neq 1}\big{|}z_{p}-z_{1}\big{|}$,
following Belavin:1984vu . Here, we have collected only those OPEs that are
expressed as Laurent series in the holomorphic variable $z$ (or
$\tilde{z}=z-j\epsilon t$) but are (anti-)holomorphic in $\bar{z}$.

The OPEs with the opposite pole-structures are noted below, with
$\Delta\tilde{\bar{z}}_{p}:=\bar{z}-\bar{z}_{p}-j\epsilon(t-t_{p})$, :

$\displaystyle\bar{P}(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left(\frac{\partial_{t_{p}}}{\Delta\tilde{\bar{z}}_{p}}+\frac{\bar{\bm{\xi}}_{p}}{(\Delta\tilde{\bar{z}}_{p})^{2}}\right)\Phi(\mathbf{x}_{p})$
(114) $\displaystyle
S_{0}^{-}(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left(\frac{z-z_{p}}{\Delta\tilde{\bar{z}}_{p}}\partial_{t_{p}}+\frac{z-z_{p}}{(\Delta\tilde{\bar{z}}_{p})^{2}}\bar{\bm{\xi}}_{p}-\frac{{\bm{\xi}}_{p}}{\Delta\tilde{\bar{z}}_{p}}\right)\Phi(\mathbf{x}_{p})$
(115)
$\displaystyle\bar{T}(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{\bar{h}_{p}}{(\Delta\tilde{\bar{z}}_{p})^{2}}+\frac{\partial_{\bar{z}_{p}}}{\Delta\tilde{\bar{z}}_{p}}-\frac{t-t_{p}}{2}\left\\{\frac{\partial_{t_{p}}}{(\Delta\tilde{\bar{z}}_{p})^{2}}+\frac{2\bar{\bm{\xi}}_{p}}{{(\Delta\tilde{\bar{z}}_{p})}^{3}}\right\\}\right]\Phi(\mathbf{x}_{p})$
(116) $\displaystyle
S_{1}^{-}(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-\frac{i}{2}\left[\frac{(z-z_{p})^{2}}{\Delta\tilde{\bar{z}}_{p}}\partial_{z_{p}}-2h_{p}\frac{z-z_{p}}{\Delta\tilde{\bar{z}}_{p}}+\frac{0}{\Delta\tilde{\bar{z}}_{p}}\right.$
$\displaystyle\hskip
122.34685pt\left.+(t-t_{p})\left(\frac{z-z_{p}}{\Delta\tilde{\bar{z}}_{p}}\partial_{t_{p}}+\frac{z-z_{p}}{(\Delta\tilde{\bar{z}}_{p})^{2}}\bar{\bm{\xi}}_{p}-\frac{{\bm{\xi}}_{p}}{\Delta\tilde{\bar{z}}_{p}}\right)\right]\Phi(\mathbf{x}_{p})$
(117)

where $\sim$ denotes ‘modulo terms (anti-)holomorphic (regular) in
$\Delta\tilde{\bar{z}}_{p}$ ’.

The Carrollian conformal primary fields can be alternatively defined as the
fields that, in the above described OPEs, have no higher order poles than
those shown explicitly and also have vanishing coefficients for the
$\frac{{(\bar{z}-\bar{z}_{p})}^{0}}{\Delta\tilde{z}_{p}}$ and
$\frac{{(z-z_{p})}^{0}}{\Delta\tilde{\bar{z}}_{p}}$ terms in the OPEs with
$S_{1}^{+}$ and $S_{1}^{-}$ respectively161616When we perform the 2D shadow
transformation e.g. (76) to reach the $T\Phi$ OPE from the $S_{1}^{-}\Phi$
OPE, the $\frac{{(z-z_{p})}^{0}}{\Delta\tilde{\bar{z}}_{p}}$ singularity gives
rise to a cubic pole in the $T_{e}\Phi$ OPE. Thus, to respect the primary OPE
(112), its coefficient must vanish.. These coefficients will be non-zero in
addition to the appearance of higher order poles in the above OPEs
corresponding to a general Carrollian conformal field.

Recalling that an OPE in a translation-invariant theory has the following
general form:

$\displaystyle{\Phi}_{1}(t_{1},\vec{x}_{1}){\Phi}_{2}(t_{2},\vec{x}_{2})=\sum_{k}C^{k}_{12}(t_{12},\vec{x}_{12})\text{
}{\Phi}_{k}(t_{2},\vec{x}_{2})$

with $\Phi_{1}$, $\Phi_{2}$ and $\Phi_{k}$ being local fields, we now extend
the above OPEs to the case of a general (non-primary) Carrollian conformal
field:

$\displaystyle
P(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\sum\limits_{n\geq
1}\frac{\left(P_{n,-1}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+2}}+\frac{\partial_{t_{p}}\Phi}{(\Delta\tilde{z}_{p})}+\frac{\bm{\xi}_{p}\cdot\Phi}{{(\Delta\tilde{z}_{p})}^{2}}\right](\mathbf{x}_{p})$
(118) $\displaystyle
S^{+}_{0}(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[(\bar{z}-\bar{z}_{p})\left(\sum\limits_{n\geq
1}\frac{\left(P_{n,-1}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+2}}+\frac{\partial_{t_{p}}\Phi}{(\Delta\tilde{z}_{p})}+\frac{\bm{\xi}_{p}\cdot\Phi}{{(\Delta\tilde{z}_{p})}^{2}}\right)\right.$
$\displaystyle\hskip 221.93149pt\left.-\left(\sum\limits_{n\geq
0}\frac{\left(P_{n,0}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+2}}+\frac{\bar{{\bm{\xi}}}_{p}\cdot\Phi}{(\Delta\tilde{z}_{p})}\right)\right](\mathbf{x}_{p})$
(119) $\displaystyle
T(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\sum\limits_{n\geq
1}\frac{\left(L_{n}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+2}}+\frac{h_{p}\Phi}{(\Delta\tilde{z}_{p})^{2}}+\frac{\partial_{z_{p}}\Phi}{\Delta\tilde{z}_{p}}\right.$
$\displaystyle\hskip
116.6563pt\left.-\frac{t-t_{p}}{2}\left\\{\sum\limits_{n\geq
1}(n+2)\frac{\left(P_{n,-1}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+3}}+\frac{\partial_{t_{p}}\Phi}{(\Delta\tilde{z}_{p})^{2}}+\frac{2\bm{\xi}_{p}\cdot\Phi}{{(\Delta\tilde{z}_{p})}^{3}}\right\\}\right](\mathbf{x}_{p})$
(120)
$\displaystyle\left(S_{1}^{+}-\frac{t-t_{p}}{2}S^{+}_{0}\right)(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-\frac{i}{2}\left[(\bar{z}-\bar{z}_{p})^{2}\left(\sum\limits_{n\geq
1}\frac{\left(j^{(-)}_{n}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+1}}+\frac{\partial_{\bar{z}_{p}}\Phi}{\Delta\tilde{z}_{p}}\right)+\sum\limits_{n\geq
0}\frac{\left(j^{(+)}_{n}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+1}}\right.$
$\displaystyle\hskip
193.47873pt\left.-2(\bar{z}-\bar{z}_{p})\left(\sum\limits_{n\geq
1}\frac{\left(j^{(0)}_{n}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+1}}+\frac{\bar{h}_{p}\Phi}{\Delta\tilde{z}_{p}}\right)\right](\mathbf{x}_{p})$
(121)

The OPE coefficients were labeled in the way that resembles the actions of
classical generators at the origin, e.g.
$\left(P_{0,-1}\Phi\right)=\bm{\xi}_{p}\cdot\Phi$ ,
$\left(P_{-1,0}\Phi\right)=\bar{\bm{\xi}}_{p}\cdot\Phi$ ,
$\left(L_{0}\Phi\right)=h_{p}\Phi$ etc. The coefficients of the $S^{+}_{1}$
OPE are labeled keeping in mind that its holomorphic weight is $h=1$.

The OPEs for the fields $T_{e}$, $j^{a}_{e}$, $P_{0}$ and $P_{-1}$ that are
easily derived from above are now listed below:

$\displaystyle
P_{-1}(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\sum\limits_{n\geq
1}\frac{\left(P_{n,-1}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+2}}+\frac{\partial_{t_{p}}\Phi}{(\Delta\tilde{z}_{p})}+\frac{\bm{\xi}_{p}\cdot\Phi}{{(\Delta\tilde{z}_{p})}^{2}}\right](\mathbf{x}_{p})$
(122) $\displaystyle
P_{0}(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\sum\limits_{n\geq-1}\frac{\left(P_{n,0}\Phi\right)+\bar{z}_{p}\left(P_{n,-1}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+2}}\right](\mathbf{x}_{p})$
(123) $\displaystyle
T_{e}(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\sum\limits_{n\geq-1}\frac{\left(L_{n}\Phi\right)+\frac{n+1}{2}t_{p}\left(P_{n-1,-1}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+2}}\right](\mathbf{x}_{p})$
(124) $\displaystyle
j^{(-)}_{e}(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-\frac{i}{2}\left[\sum\limits_{n\geq
1}\frac{\left(j^{(-)}_{n}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+1}}+\frac{\partial_{\bar{z}_{p}}\Phi}{\Delta\tilde{z}_{p}}\right](\mathbf{x}_{p})$
(125) $\displaystyle
j^{(0)}_{e}(t,z,\bar{z})\Phi(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-\frac{i}{2}\left[\sum\limits_{n\geq
1}\frac{\left(j^{(0)}_{n}\Phi\right)+\bar{z}_{p}\left(j^{(-)}_{n}\Phi\right)+\frac{t_{p}}{2}\left(P_{n-1,-1}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+1}}\right.$
$\displaystyle\hskip
236.15787pt\left.+\frac{\left(\bar{z}_{p}\partial_{\bar{z}_{p}}+\frac{t_{p}}{2}\partial_{t_{p}}+\bar{h}_{p}\right)\Phi}{\Delta\tilde{z}_{p}}\right](\mathbf{x}_{p})$
(126) $\displaystyle j^{(+)}_{e}\Phi\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-\frac{i}{2}\left[\sum\limits_{n\geq
1}\frac{\left(j^{(+)}_{n}\Phi\right)+2\bar{z}_{p}\left(j^{(0)}_{n}\Phi\right)+\bar{z}^{2}_{p}\left(j^{(-)}_{n}\Phi\right)+t_{p}\bar{z}_{p}\left(P_{n-1,-1}\Phi\right)+t_{p}\left(P_{n-1,0}\Phi\right)}{{(\Delta\tilde{z}_{p})}^{n+1}}\right.$
$\displaystyle\hskip
113.81102pt\left.+\frac{\left(\bar{z}^{2}_{p}\partial_{\bar{z}_{p}}+\bar{z}_{p}t_{p}\partial_{t_{p}}+2\bar{z}_{p}\bar{h}_{p}\right)\Phi+t_{p}\left(\bar{{\bm{\xi}}}_{p}\cdot\Phi\right)+\left(j^{(+)}_{0}\Phi\right)}{\Delta\tilde{z}_{p}}\right](\mathbf{x}_{p})$
(127)

We recognize that the $j^{a}_{e}\Phi$ OPEs looks exactly the same as the
holomorphic Kac-Moody current OPEs while the $T_{e}\Phi$ OPE resembles a
holomorphic Virasoro OPE. The interpretation of the $j^{a}_{e}$ fields as the
generators of a Kac-Moody symmetry Banerjee:2020zlg ; Guevara:2021abz ;
Polyakov:1987zb  is further hinted by the fact that each of them has $h=1$. On
the other hand, the fact that $T_{e}$ has $h=2$ hints toward an interpretation
of the $T_{e}$ field as a Virasoro energy-momentum tensor Kapec:2016jld ;
Cheung:2016iub . But these interpretations are not straight-forward, since
some of these fields have non-zero $\bar{h}$ unlike in 2D holomorphic CFTs.
Nevertheless, below we shall proceed to define the modes of these fields and
to find their algebra.

Similar OPEs can be obtained for the anti-holomorphic sector.

###  5.2 The modes and their actions

It is to be noted that these six fields are time-independent and holomorphic
when written in an OPE. Thus, they can be mode-expanded (valid only in OPEs)
as below:

$\displaystyle T_{e}(z)=\sum\limits_{n\in\mathbb{Z}}L_{n}z^{-n-2}\hskip
7.11317pt;\hskip 7.11317ptL_{n}=\frac{1}{2\pi
j}\oint\limits_{C_{u}^{\prime}}d\hat{z}\text{ }\hat{z}^{n+1}\text{
}T_{e}(t,\hat{z},\hat{\bar{z}})=\frac{1}{2\pi i}\oint\limits_{C_{e}}dz\text{
}z^{n+1}\text{ }T_{e}(t,z,\bar{z})$ $\displaystyle
j^{a}_{e}(z)=\frac{1}{2}\sum\limits_{n\in\mathbb{Z}}j^{a}_{n}z^{-n-1}\hskip
7.11317pt;\hskip 7.11317ptj^{a}_{n}=\frac{2}{2\pi
j}\oint\limits_{C_{u}^{\prime}}d\hat{z}\text{ }\hat{z}^{n}\text{
}j^{a}_{e}(t,\hat{z},\hat{\bar{z}})=\frac{2}{2\pi
i}\oint\limits_{C_{e}}dz\text{ }z^{n}\text{ }j^{a}_{e}(t,z,\bar{z})$ (128)
$\displaystyle P_{i}(z)=\sum\limits_{n\in\mathbb{Z}}P_{n,i}z^{-n-2}\hskip
7.11317pt;\hskip 7.11317ptP_{n,i}=\frac{1}{2\pi
j}\oint\limits_{C_{u}^{\prime}}d\hat{z}\text{ }\hat{z}^{n+1}\text{
}P_{i}(t,\hat{z},\hat{\bar{z}})=\frac{1}{2\pi i}\oint\limits_{C_{e}}dz\text{
}z^{n+1}\text{ }P_{i}(t,z,\bar{z})$ (129)

with $i\in\\{0,-1\\}$.

We first note the actions of the zero-modes $j^{a}_{0}$ on an arbitrary field;
this can be obtained171717Section 5.3 contains an illustration. from the
analogue of (100) applied on the OPEs (125)-(127):

$\displaystyle\left[j_{0}^{(-)}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\partial_{\bar{z}_{p}}\Phi(\mathbf{x}_{p})\hskip
14.22636pt;\hskip 14.22636pt\left[j_{0}^{(0)}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\left[\bar{z}_{p}\partial_{\bar{z}_{p}}+\frac{t_{p}}{2}\partial_{t_{p}}+\bar{h}_{p}\right]\Phi(\mathbf{x}_{p})$
$\displaystyle\left[j_{0}^{(+)}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\left[\left(\bar{z}^{2}_{p}\partial_{\bar{z}_{p}}+\bar{z}_{p}t_{p}\partial_{t_{p}}+2\bar{z}_{p}\bar{h}_{p}\right)\Phi+t_{p}\left(\bar{{\bm{\xi}}}_{p}\cdot\Phi\right)+\left(j^{(+)}_{0}\Phi\right)\right](\mathbf{x}_{p})$
(130)

Comparing these actions with (23) and (27) for a general non-multiplet field
(i.e. fields with $\bar{\bm{\mathcal{L}}}_{1}(\mathbf{0})\Phi(\mathbf{0})\neq
0$), we see that the zero-modes of the three $j^{a}_{e}$ fields generate the
three anti-holomorphic Lorentz, i.e. the $\overline{\text{SL}(2,\mathbb{R})}$
transformations on the space of quantum fields. Thus, the possible Kac-Moody
algebra generated by the $j^{a}_{e}$ fields can be the
$\overline{\text{sl}(2,\mathbb{R})}$ current algebra.

On the other hand, from the OPE (124), we find that the three modes $L_{n}$
with $n\in\\{0,\pm 1\\}$ generate the three holomorphic Lorentz, i.e. the
$\text{SL}(2,\mathbb{R})$ transformations of the quantum fields:

$\displaystyle\left[L_{-1}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\partial_{z_{p}}\Phi(\mathbf{x}_{p})\hskip
14.22636pt;\hskip 14.22636pt\left[L_{0}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\left[z_{p}\partial_{z_{p}}+\frac{t_{p}}{2}\partial_{t_{p}}+h_{p}\right]\Phi(\mathbf{x}_{p})$
$\displaystyle\left[L_{1}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\left[\left(z_{p}^{2}\partial_{z_{p}}+z_{p}t_{p}\partial_{t_{p}}+2z_{p}h_{p}\right)\Phi+t_{p}\left(\bm{\xi}_{p}\cdot\Phi\right)+\left(L_{1}\Phi\right)\right](\mathbf{x}_{p})$
(131)

This was expected from the definition (97) of the quantum holomorphic super-
rotation generators.

Thus, we see that the six modes $L_{n}$ with $n\in\\{0,\pm 1\\}$ and
$j^{a}_{0}$ together generate the Lorentz i.e. $\text{SL}(2,\mathbb{C})$
transformations of the quantum fields in $1+2$D. In addition to that, from the
OPEs (122) and (123), we find that the four modes $P_{a,b}$ with
$a,b\in\\{-1,0\\}$ implement the four translations of the Poincare group
$\text{ISL}(2,\mathbb{C})$ on the $1+2$D quantum fields:

$\displaystyle\left[P_{-1,-1}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\partial_{t_{p}}\Phi(\mathbf{x}_{p})$
$\displaystyle\left[P_{0,-1}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\left[z_{p}\partial_{t_{p}}+\bm{\xi}_{p}\right]\Phi(\mathbf{x}_{p})\hskip
14.22636pt;\hskip 14.22636pt\left[P_{-1,0}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\left[\bar{z}_{p}\partial_{t_{p}}+\bar{\bm{\xi}}_{p}\right]\Phi(\mathbf{x}_{p})$
(132) $\displaystyle\left[P_{0,0}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\left[z_{p}\bar{z}_{p}\partial_{t_{p}}\Phi+z_{p}\left(\bar{\bm{\xi}}_{p}\cdot\Phi\right)+\bar{z}_{p}\left({\bm{\xi}}_{p}\cdot\Phi\right)+\left(P_{0,0}\Phi\right)\right](\mathbf{x}_{p})$

We shall now list the actions of the modes on a $1+2$D CC primary field that
are derived from the primary OPEs (with $n\in\mathbb{Z}$):

$\displaystyle\left[P_{n,-1}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\left[z_{p}^{n+1}\partial_{t_{p}}+(n+1)z_{p}^{n}\bm{\xi}_{p}\right]\Phi(\mathbf{x}_{p})$
$\displaystyle\left[P_{n,0}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\left[z_{p}^{n+1}\left(\bar{z}_{p}\partial_{t_{p}}+\bar{\bm{\xi}}_{p}\right)+(n+1)z_{p}^{n}\bar{z}_{p}\bm{\xi}_{p}\right]\Phi(\mathbf{x}_{p})$
(133) $\displaystyle\left[L_{n}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\left[z_{p}^{n+1}\partial_{z_{p}}+\frac{n+1}{2}z_{p}^{n}t_{p}\partial_{t_{p}}+(n+1)z_{p}^{n}h_{p}+(n+1)nz_{p}^{n-1}\frac{t_{p}}{2}\bm{\xi}_{p}\right]\Phi(\mathbf{x}_{p})$
(134) $\displaystyle\left[j^{(-)}_{n}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-iz_{p}^{n}\partial_{\bar{z}_{p}}\Phi(\mathbf{x}_{p})$
$\displaystyle\left[j^{(0)}_{n}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\left[z^{n}_{p}\left(\bar{z}_{p}\partial_{\bar{z}_{p}}+\frac{t_{p}}{2}\partial_{t_{p}}+\bar{h}_{p}\right)+nz^{n-1}_{p}\frac{t_{p}}{2}\bm{\xi}_{p}\right]\Phi(\mathbf{x}_{p})$
(135) $\displaystyle\left[j^{(+)}_{n}\text{ },\text{
}\Phi(\mathbf{x}_{p})\right]=-i\left[z_{p}^{n}\left(\bar{z}^{2}_{p}\partial_{\bar{z}_{p}}+\bar{z}_{p}t_{p}\partial_{t_{p}}+2\bar{z}_{p}\bar{h}_{p}+t_{p}\bar{{\bm{\xi}}}_{p}\right)+nz_{p}^{n-1}\bar{z}_{p}{t_{p}}\bm{\xi}_{p}\right]\Phi(\mathbf{x}_{p})$

While the first three are recognized respectively to be the $z^{n+1}$ and
$z^{n+1}\bar{z}$ super-translations and holomorphic super-rotations from (28)
and (29), the classical counterparts of the last three transformations were
not discussed. They can be thought of as the infinitesimal versions of the
following Carrollian diffeomorphisms, with $q=-1,0,1$ respectively, on the
$1+2$D Carrollian space-time:

$\displaystyle z\rightarrow z^{\prime}=z\hskip 7.11317pt;\hskip
7.11317pt\bar{z}\rightarrow\bar{z}^{\prime}=\bar{z}+\bar{z}^{q+1}f(z)\hskip
7.11317pt;\hskip 7.11317ptt\rightarrow
t^{\prime}=t{[1+(q+1)\bar{z}^{q}f(z)]}^{\frac{1}{2}}$ (136)

with $f(z)$ being a meromorphic function. They generalize the $S^{2}$
diffeomorphisms discussed in Campiglia:2014yka ; Campiglia:2015yka ;
Compere:2018ylh ; Donnay:2020guq ; Campiglia:2020qvc .

From the actions of the modes $j^{(+)}_{0}$ and $L_{1}$, it is clear that if a
field $\Phi$ is to transform covariantly (i.e. like the CC primary fields)
under the Lorentz i.e. $\text{SL}(2,\mathbb{C})$ transformations, it must
have:

$\displaystyle\left(j^{(+)}_{0}\Phi\right)=0=\left(L_{1}\Phi\right)$ (137)

These criteria are equivalent to the definition of quasi-primary fields in 2D
CFTs Belavin:1984vu ; these fields will be called ‘Lorentz quasi-primaries’.
If, in addition to these, $\left(P_{0,0}\Phi\right)=0$ holds the field
covariantly transforms under the Poincare i.e. $\text{ISL}(2,\mathbb{C})$
group. The quantum field is then called a $1+2$D CC quasi-primary field; thus,
it transforms covariantly under the BMS4 transformations globally defined on
$\mathbb{R}\times S^{2}$.

Thus from the holomorphic OPEs, we obtain all of the holomorphic super-
rotations but only two types of super-translations and the quantum charges
(modes) that generate those. This is consistent with our discussion in section
4.2 where we encountered various difficulties in defining finite charges
generating the complete set of the super-translations. Seemingly as a
compensation to this, the holomorphic OPEs allowed us to define charges
generating a special kind of Carrollian diffeomorphism (136) that is not a
subset of the (extended) BMS4 transformations.

Similarly, the anti-holomorphic OPEs give all the anti-holomorphic super-
rotations generated by $\bar{L}_{n}$ but only two types of super-translations
generated $P_{-1,n}$ and $P_{0,n}$ that are conjugate to those obtained in the
holomorphic case along with the anti-holomorphic version of the diffeomorphism
(136). Here, the zero-modes of the three Kac-Moody currents generate the
$\text{SL}(2,\mathbb{R})$ transformations while $\bar{L}_{n}$ with
$n\in\\{0,\pm 1\\}$ generate the $\overline{\text{SL}(2,\mathbb{R})}$
transformations; thus, together they generate the $\text{SL}(2,\mathbb{C})$
transformations.

But, we can not conclude that we have obtained a union of the two sets of
quantum transformations coming from the OPEs of the two sectors. This is so
because the union of the corresponding quantum generators (modes) does not
form an algebra, as we shall now see.

###  5.3 The symmetry algebra

We begin by recalling that all the field-operators involved in an OPE must be
local fields. So, a field and its shadow transformation which is a non-local
object can not both be involved in an OPE Banerjee:2022wht . Below we shall
explicitly see the problems that arise in the case of $1+2$D CCFT when we try
to construct an OPE of a field and its shadow.

The seemingly infinite Laurent series in the holomorphic sector OPEs can be
truncated at finite order poles by demanding that there be no local field with
negative holomorphic weight181818Following Banerjee:2022wht , this assumption
may be interpreted as to be that the Carrollian conformal fields with $h<0$
are to be treated as the non-local 2D shadow-transformations of the $h>1$
fields which are taken to be the local fields in the holomorphic sector. i.e.
all the local fields have $h\geq 0$. In the anti-holomorphic sector, the
similar demand would be $\bar{h}\geq 0$ for all local fields. One of these
will be a very crucial assumption in what follows, similar to the situation in
a 2D relativistic CFT Zamolodchikov:1989mz . Both of these demands are more
relaxing than the demand that scaling dimensions $\Delta\geq 0$ for all local
fields (in any sector).

Following the celestial CFT literature e.g. Fotopoulos:2019tpe , we shall also
assume that the eight fields $P,\bar{P},T,\bar{T},S^{\pm}_{0}$ and
$S^{\pm}_{1}$ either transform covariantly under the Lorentz i.e.
$\text{SL}(2,\mathbb{C})$ transformations (i.e. either they are Lorentz quasi-
primaries) or are global descendants of appropriate Lorentz covariant fields.
We need not bother at this stage if they are $1+2$D CC quasi-primaries or
descendants thereof.

As will be demonstrated, these two assumptions stated above together with the
bosonic exchange property of these fields (all of them have integer spins) are
strong enough to completely specify the pole-singularities of the (allowed)
mutual OPEs of those eight fields. From these pole structures, we can readily
deduce the algebra of the modes.

We now have to choose the sets of the local fields from those eight fields
such that members of one set can consistently form mutual OPEs. From the
definitions in section 3, it is evident that only one member from each of the
following sets can be a local field: $\\{P,\bar{P}\\}$,
$\\{S^{+}_{0},S^{-}_{0}\\}$, $\\{T,S^{-}_{1}\\}$ and
$\\{S^{+}_{1},\bar{T}\\}$. We discuss on the following remaining
possibilities:

  1. 1.

Taking $P$ as a local Lorentz quasi-primary field: This choice immediately
tells that $S^{+}_{0}$ can not be a Lorentz quasi-primary but still be a local
field while $\bar{P}$ and $S^{-}_{0}$ can not even be local fields.

We recall that, in the derivation of (104), the contact term (on $S^{2}$) in
the primary Ward identity (52) played a very important role; due to this term,
(104) remains valid for arbitrary super-translations. But, as discussed in the
beginning of this section, such a contact term does not appear in the OPEs.
Hence this choice makes us lose crucial information when we go from Ward
identity to the OPE.

Besides, we expect from (118) that any general OPE191919This is due to the
fact that the OPE must be associative. Along with the two local fields which
are explicitly shown in the L.H.S. of an OPE, there are $n(\geq 0)$ other
local fields. We can choose any pairs from this $n+2$ local fields and use the
OPEs corresponding to those pairs. All of these different pairings (and OPEs)
should give the same final result: this is the statement of the crossing
symmetry hypothesis. with $\bar{\partial}P$ on the L.H.S. is regular (up to
contact terms). But the $S^{+}_{1}\bar{\partial}P$ OPE violates this: since
$P$ has non-vanishing $\bar{h}=\frac{1}{2}$ and it is assumed to be a Lorentz
quasi-primary, we have a simple pole singularity in the
$S^{+}_{1}\bar{\partial}P$ OPE coming from (121) that does not go away even if
we assume $P$ to be a primary field. Similarly, the $\bar{T}\bar{\partial}P$
and the $S^{-}_{1}\bar{\partial}P$ OPEs contain pole singularities instead of
being regular.

Thus, if $P$ is to be taken as a Lorentz quasi-primary local field, both
$S_{1}^{\pm}$ and $\bar{T}$ can not be local fields. But the full
$\text{isl}(2,\mathbb{C})$ will then not be a sub-algebra of the resulting
mode-algebra because all of the $\overline{\text{sl}(2,\mathbb{R})}$
generators will be absent.

Hence this is not a valid choice i.e. $P$ (or $\bar{P}$) can not be treated as
a Lorentz quasi-primary field.

  2. 2.

Taking $S^{+}_{0}$ as a local Lorentz quasi-primary field: This immediately
renders $S^{-}_{0}$ and $\bar{P}$ non-local while $P$ is now a local but not
Lorentz quasi-primary descendant of $S^{+}_{0}$. Also, the $S_{0}^{+}$ Ward
identities (for primary fields) do not contain any contact terms, so we can
directly write the corresponding OPEs without any loss.

Now, any general OPE with $\bar{\partial}^{2}S^{+}_{0}$ on the L.H.S. should
be regular (up to contact terms), following from (119). But
$\bar{T}\bar{\partial}^{2}S^{+}_{0}$ and
$S_{1}^{-}\bar{\partial}^{2}S^{+}_{0}$ OPEs both have pole singularities that
persist even if $S^{+}_{0}$ is assumed to be a primary field. This happens as
both $h,\bar{h}$ are non-zero for $S^{+}_{0}$. Thus, under this choice
$\bar{T}$ and $S^{-}_{1}$ can not be treated as local fields.

On the other hand, we can not yet conclude if both
${T}\bar{\partial}^{2}S^{+}_{0}$ and $S_{1}^{+}\bar{\partial}^{2}S^{+}_{0}$
are regular. So we can proceed assuming for the time being that $T$,
$S_{1}^{+}$ and $S_{0}^{+}$ can be simultaneously treated as local Lorentz
quasi-primary fields and we will find that this is indeed the case.

  3. 3.

Taking $S^{-}_{0}$ as a local Lorentz quasi-primary field: Similarly, in this
case ${T}$, $S^{+}_{1}$, $P$ and $S^{+}_{0}$ can not be treated as local
fields.

  4. 4.

Taking both $T$ and $\bar{T}$ as local Lorentz quasi-primary fields: Here both
$S^{\pm}_{1}$ are to be treated as non-local shadows. But from the discussion
similar to the first two cases, it is evident that none of $P$, $\bar{P}$ and
$S^{\pm}_{0}$ can form consistent OPEs with both of $T$ and $\bar{T}$. From
(120), any OPE with $\bar{\partial}T$ or $\partial\bar{T}$ on the L.H.S. is
expected to be regular; but the OPEs $S^{+}_{0}\partial\bar{T}$ (hence
$P\partial\bar{T}$) and $S^{-}_{0}\bar{\partial}T$ (hence
$\bar{P}\bar{\partial}T$) all have pole singularities. Thus, only $T$ and
$\bar{T}$ can be treated as local fields in this case.

Also, from the definition (70), it follows that both $\partial P$ and
$\bar{\partial}\bar{P}$ have to be local fields in order to both $T$ and
$\bar{T}$ being local. Clearly, $\partial P$ and $\bar{\partial}\bar{P}$ both
can not be local simultaneously.

Moreover, simultaneously treating both $T_{e}$ and $\bar{T}_{e}$ local is also
unacceptable since all the super-translation generators then will be absent
from the resulting mode-algebra implying that the $\text{isl}(2,\mathbb{C})$
is not even a sub-algebra of the same.

  5. 5.

Taking both $S^{\pm}_{1}$ as local Lorentz quasi-primary fields: This is not
possible. A general OPE with $\bar{\partial}^{3}S_{1}^{+}$ (or
${\partial}^{3}S_{1}^{-}$) on the L.H.S. is expected to be regular, as seen
from (121). This is not what happens to the
$S_{1}^{-}\bar{\partial}^{3}S_{1}^{+}$ OPE which has anti-meromorphic pole
singularities. On the other hand, the OPE $S_{1}^{+}{\partial}^{3}S_{1}^{-}$
contains meromorphic poles instead of being regular. Hence, both $S^{\pm}_{1}$
can not be treated as local fields simultaneously.

Therefore we have to pick one among the two sectors to form consistent mutual
OPEs Banerjee:2022wht . We proceed with the holomorphic one by treating the
fields $T$, $S^{+}_{0}$ and $S^{+}_{1}$ as local Lorentz quasi-primary fields
while demoting the others to mere non-local shadows or descendants thereof.
This is the holomorphic sector of the $1+2$D Carrollian CFT. Thus, we shall
also assume that all the local fields in the theory have holomorphic weights
$h\geq 0$.

Obeying the general form (119) of the $S^{+}_{0}$ OPE and being consistent
with the two assumptions, we write down the following ansatz for the
$S^{+}_{0}S^{+}_{0}$ OPE:

$\displaystyle
S^{+}_{0}(t,z,\bar{z})S^{+}_{0}(t_{p},z_{p},\bar{z}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[(\bar{z}-\bar{z}_{p})\left(\frac{\left(P_{1,-1}S^{+}_{0}\right)}{{(\Delta\tilde{z}_{p})}^{3}}+\frac{\bm{\xi}\cdot
S^{+}_{0}}{{(\Delta\tilde{z}_{p})}^{2}}\right)\right.$ $\displaystyle\hskip
170.71652pt\left.-\left(\frac{\left(P_{1,0}S^{+}_{0}\right)}{{(\Delta\tilde{z}_{p})}^{3}}+\frac{\left(P_{0,0}S^{+}_{0}\right)}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{\bar{{\bm{\xi}}}\cdot
S^{+}_{0}}{(\Delta\tilde{z}_{p})}\right)\right](t_{p},z_{p},\bar{z}_{p})$
(138)

Because $\partial_{t}S^{+}_{0}\sim 0$ within a correlator (or an OPE), all the
local fields appearing on the R.H.S. have this property in common. Also, the
condition
$S^{+}_{0}(t,z,\bar{z})\bar{\partial}^{2}_{p}S^{+}_{0}(t_{p},z_{p},\bar{z}_{p})\sim
0$ gives rise to some constraints between these fields.

Moreover, since $S^{+}_{0}$ has integer spin $m=2$, it should obey the bosonic
exchange property, i.e. we expect as an OPE statement that:

$\displaystyle S^{+}_{0}(t,z,\bar{z})S^{+}_{0}(t_{p},z_{p},\bar{z}_{p})\sim
S^{+}_{0}(t_{p},z_{p},\bar{z}_{p})S^{+}_{0}(t,z,\bar{z})$

We first expand the OPE on the R.H.S. as:

$\displaystyle
S^{+}_{0}(t_{p},z_{p},\bar{z}_{p})S^{+}_{0}(t,z,\bar{z})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[(\bar{z}-\bar{z}_{p})\left(\frac{\left(P_{1,-1}S^{+}_{0}\right)}{{(\Delta\tilde{z}_{p})}^{3}}-\frac{\bm{\xi}\cdot
S^{+}_{0}}{{(\Delta\tilde{z}_{p})}^{2}}\right)\right.$ $\displaystyle\hskip
170.71652pt\left.+\left(\frac{\left(P_{1,0}S^{+}_{0}\right)}{{(\Delta\tilde{z}_{p})}^{3}}-\frac{\left(P_{0,0}S^{+}_{0}\right)}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{\bar{{\bm{\xi}}}\cdot
S^{+}_{0}}{(\Delta\tilde{z}_{p})}\right)\right](t,z,\bar{z})$ (139)

For comparing the both sides of the bosonic exchange property, we now need to
perform a multi-variate Taylor expansion of the fields at $(t,z,\bar{z})$
around $(t_{p},z_{p},\bar{z}_{p})$ on the R.H.S. of (139) and look for order-
by-order matching of the coefficients of
$\frac{(\bar{z}-\bar{z}_{p})^{s}}{(z-z_{p})^{r}}$ , where $r\geq 1$ and $s\geq
0$, with those on the R.H.S. of (138).

E.g. we can promptly notice that $\left(P_{1,0}S^{+}_{0}\right)\sim 0$ i.e.
$\left(P_{1,0}S^{+}_{0}\right)$ is $0$ upto contact terms inside a correlator.
The constraints coming from the bosonic exchange property and from the
requirement that
$S^{+}_{0}(t,z,\bar{z})\bar{\partial}^{2}_{p}S^{+}_{0}(t_{p},z_{p},\bar{z}_{p})\sim
0$ have some overlap.

Next we make an ansatz for the $S_{1}^{+}S^{+}_{0}$ OPE consistent with the
assumptions202020We recall from (137) that if $S^{+}_{0}$ is to be a Lorentz
quasi-primary it must satisfy: $\left(j^{(+)}_{0}S^{+}_{0}\right)=0$. and the
fact that $S^{+}_{0}$ has $\bar{h}=-\frac{1}{2}$, looking at the general form
(121):

$\displaystyle
S_{1}^{+}(t,z,\bar{z})S^{+}_{0}(t_{p},z_{p},\bar{z}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-\frac{i}{2}\left[(\bar{z}-\bar{z}_{p})^{2}\left(\frac{\Phi_{\frac{1}{2},\frac{1}{2}}}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{\partial_{\bar{z}_{p}}S^{+}_{0}}{\Delta\tilde{z}_{p}}\right)+(\bar{z}-\bar{z}_{p})\left(\frac{\Phi_{\frac{1}{2},-\frac{1}{2}}}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{S^{+}_{0}}{\Delta\tilde{z}_{p}}\right)\right.$
$\displaystyle\left.+\frac{\Phi_{\frac{1}{2},-\frac{3}{2}}}{{(\Delta\tilde{z}_{p})}^{2}}+(t-t_{p})\left\\{(\bar{z}-\bar{z}_{p})\left(\frac{\left(P_{1,-1}S^{+}_{0}\right)}{{(\Delta\tilde{z}_{p})}^{3}}+\frac{\bm{\xi}\cdot
S^{+}_{0}}{{(\Delta\tilde{z}_{p})}^{2}}\right)-\left(\frac{\left(P_{0,0}S^{+}_{0}\right)}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{\bar{{\bm{\xi}}}\cdot
S^{+}_{0}}{(\Delta\tilde{z}_{p})}\right)\right\\}\right](\mathbf{x}_{p})$
(140)

where the $\Phi_{h,\bar{h}}$ are as of yet undetermined local fields.

On the other hand, following from (119), the ansatz for the
$S_{0}^{+}S^{+}_{1}$ OPE is allowed to be:

$\displaystyle
S^{+}_{0}(t,z,\bar{z})S^{+}_{1}(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[(\bar{z}-\bar{z}_{p})\left(\frac{\bm{\xi}\cdot
S^{+}_{1}}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{\partial_{t_{p}}S^{+}_{1}}{{(\Delta\tilde{z}_{p})}}\right)+\frac{\frac{1}{2}\Phi_{\frac{1}{2},-\frac{3}{2}}}{{(\Delta\tilde{z}_{p})}^{2}}-\frac{\bar{{\bm{\xi}}}\cdot
S^{+}_{1}}{(\Delta\tilde{z}_{p})}\right](\mathbf{x}_{p})$ (141)

Now, since both of $S^{+}_{1}$ and $S^{+}_{0}$ has integer spin $m=2$, they
satisfy the following bosonic exchange property:

$\displaystyle S^{+}_{1}(t,z,\bar{z})S^{+}_{0}(t_{p},z_{p},\bar{z}_{p})\sim
S^{+}_{0}(t_{p},z_{p},\bar{z}_{p})S^{+}_{1}(t,z,\bar{z})$

Thus, to compare we need to Taylor-expand the R.H.S. of the following OPE
around $(t_{p},z_{p},\bar{z}_{p})$ and compare order-by-order with the R.H.S.
of (140):

$\displaystyle
S^{+}_{0}(t_{p},z_{p},\bar{z}_{p})S^{+}_{1}(\mathbf{x})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[(\bar{z}-\bar{z}_{p})\left(-\frac{\bm{\xi}\cdot
S^{+}_{1}}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{\partial_{t_{p}}S^{+}_{1}}{{(\Delta\tilde{z}_{p})}}\right)+\frac{\frac{1}{2}\Phi_{\frac{1}{2},-\frac{3}{2}}}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{\bar{{\bm{\xi}}}\cdot
S^{+}_{1}}{(\Delta\tilde{z}_{p})}\right](t,z,\bar{z})$

Upon comparison (and using the fact that $\left(P_{0,0}S^{+}_{0}\right)$ has
$(h,\bar{h})=(1,-1)$ ) we find:

$\displaystyle\left(P_{1,-1}S^{+}_{0}\right)\sim{{\bm{\xi}}}\cdot
S^{+}_{0}\sim\bar{{\bm{\xi}}}\cdot S^{+}_{0}\sim 0$
$\displaystyle\partial_{t}\left(P_{0,0}S^{+}_{0}\right)\sim\partial_{z}\left(P_{0,0}S^{+}_{0}\right)\sim\partial_{\bar{z}}\left(P_{0,0}S^{+}_{0}\right)\sim
0\hskip 7.11317pt\Longrightarrow\hskip
7.11317pt{\left(P_{0,0}S^{+}_{0}\right)}\sim 0$
$\displaystyle\partial_{t}\left({{\bm{\xi}}}\cdot
S^{+}_{1}\right)\sim\partial_{t}\left(\bar{{\bm{\xi}}}\cdot
S^{+}_{1}\right)\sim\partial_{t}\Phi_{\frac{1}{2},-\frac{3}{2}}\sim\partial_{t}\Phi_{\frac{1}{2},\frac{1}{2}}\sim\partial_{t}\Phi_{\frac{1}{2},-\frac{1}{2}}\sim
0$

and some other not-so-simple constraints which can also be obtained from the
requirements that $S^{+}_{0}\bar{\partial}^{3}S^{+}_{1}\sim 0$ ,
$S^{+}_{1}\bar{\partial}^{2}S^{+}_{0}\sim 0$ and
$\left(\partial_{t}S^{+}_{1}-\frac{1}{2}S^{+}_{0}\right)\sim 0$. The first two
lines ensure that there are no singularity in the $S^{+}_{0}S^{+}_{0}$ OPE.

Finally, we make the ansatz for the $S^{+}_{1}S^{+}_{1}$ OPE, keeping in mind
that $S^{+}_{1}$ has $\bar{h}=-1$ and $\left(j^{(+)}_{0}S^{+}_{1}\right)=0$
due to the assumption that $S^{+}_{1}$ is Lorentz quasi-primary:

$\displaystyle
S_{1}^{+}(t,z,\bar{z})S^{+}_{1}(t_{p},z_{p},\bar{z}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-\frac{i}{2}\left[(\bar{z}-\bar{z}_{p})^{2}\left(\frac{\Phi_{0,0}}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{\partial_{\bar{z}_{p}}S^{+}_{1}}{\Delta\tilde{z}_{p}}\right)+(\bar{z}-\bar{z}_{p})\left(\frac{\Phi_{0,-1}}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{2S^{+}_{1}}{\Delta\tilde{z}_{p}}\right)\right.$
$\displaystyle\hskip
42.67912pt\left.+\frac{\Phi_{0,-2}}{{(\Delta\tilde{z}_{p})}^{2}}+(t-t_{p})\left\\{(\bar{z}-\bar{z}_{p})\left(\frac{\bm{\xi}\cdot
S^{+}_{1}}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{\partial_{t_{p}}S^{+}_{1}}{{(\Delta\tilde{z}_{p})}}\right)+\frac{\frac{1}{2}\Phi_{\frac{1}{2},-\frac{3}{2}}}{{(\Delta\tilde{z}_{p})}^{2}}-\frac{\bar{{\bm{\xi}}}\cdot
S^{+}_{1}}{(\Delta\tilde{z}_{p})}\right\\}\right](\mathbf{x}_{p})$ (142)

where the fields $\Phi_{h,\bar{h}}$ are yet undetermined local fields. Using
the bosonic exchange property similarly as in the case of $S^{+}_{0}S^{+}_{0}$
OPE, we obtain several constraints that combined with the previously obtained
ones completely determine all the $\Phi_{h,\bar{h}}$ fields (within an OPE).
The results, consistent with the dimensions of the $\Phi_{h,\bar{h}}$ fields,
are:

$\displaystyle\left({{\bm{\xi}}}\cdot
S^{+}_{1}\right)\sim\left(\bar{{\bm{\xi}}}\cdot
S^{+}_{1}\right)\sim\Phi_{\frac{1}{2},-\frac{3}{2}}\sim\Phi_{\frac{1}{2},\frac{1}{2}}\sim\Phi_{\frac{1}{2},-\frac{1}{2}}\sim
0\hskip 14.22636pt;\hskip 14.22636pt\Phi_{0,0}\sim K$
$\displaystyle\Phi_{0,-2}(\mathbf{x}_{1})\Psi(\mathbf{x}_{2})\sim
A{(\bar{z}_{1}-\bar{z}_{2})}^{2}\Psi(\mathbf{x}_{2})\hskip 14.22636pt;\hskip
14.22636pt\Phi_{0,-1}(\mathbf{x}_{1})\Psi(\mathbf{x}_{2})\sim
A{(\bar{z}_{1}-\bar{z}_{2})}\Psi(\mathbf{x}_{2})$

where $K$ and $A$ are two constants and $\Psi$ is an arbitrary local field.

Since
$\left(P_{0,0}S^{+}_{1}\right)=-\frac{1}{2}\Phi_{\frac{1}{2},-\frac{3}{2}}\sim
0$ and we assumed $S^{+}_{1}$ to be Lorentz quasi-primary, we have just shown
that $S^{+}_{1}$ is a $1+2$D CC quasi-primary field. Moreover,
${{\bm{\xi}}}\cdot S^{+}_{1}\sim\bar{{\bm{\xi}}}\cdot S^{+}_{1}\sim 0$ and, as
we shall see, $S^{+}_{1}$ does not mix with any other field under Carrollian
boost; these imply that $S^{+}_{1}$ transforms under an irrep of the spin-
boost sub-algebra (15). Thus, under the global $1+2$D CC i.e.
$\text{ISL}(2,\mathbb{C})$ transformations, $S^{+}_{1}$ transforms as a 2D CFT
quasi-primary i.e. as (32). Correlation function between two such $1+2$D CC
quasi-primaries were derived in Bagchi:2016bcd  to be exactly of the same form
as that of the 2D CFT quasi-primary two-point functions. So, the $\left\langle
S_{1}^{+}S^{+}_{1}\right\rangle$ correlator should be:

$\displaystyle\left\langle
S_{1}^{+}(t_{1},z_{1},\bar{z}_{1})S^{+}_{1}(t_{2},z_{2},\bar{z}_{2})\mathbf{I}(t_{3},z_{3},\bar{z}_{3})\right\rangle\propto\lim\limits_{\epsilon\rightarrow
0^{+}}\frac{{(\bar{z}_{1}-\bar{z}_{2})}^{2}}{\left[{(z_{1}-z_{2})-j\epsilon(t_{1}-t_{2})}\right]^{2}}$
(143)

If we calculate this correlator using the $S_{1}^{+}S^{+}_{1}$ OPE and the
fact that $\left\langle S^{+}_{1}\right\rangle$=0, it is immediately seen that
$A=0$ must hold.

Thus, we note down the final form of the following OPEs:

$\displaystyle S_{0}^{+}(\mathbf{x})S^{+}_{0}(\mathbf{x}_{p})\sim 0\hskip
14.22636pt;\hskip
14.22636ptS_{0}^{+}(\mathbf{x})S^{+}_{1}(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\text{ }\frac{\bar{z}-\bar{z}_{p}}{{(\Delta\tilde{z}_{p})}}\text{
}\partial_{t_{p}}S^{+}_{1}(\mathbf{x}_{p})$ $\displaystyle
S_{1}^{+}(\mathbf{x})S^{+}_{0}(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-\frac{i}{2}\left[\frac{(\bar{z}-\bar{z}_{p})^{2}}{{(\Delta\tilde{z}_{p})}}\text{
}\partial_{\bar{z}_{p}}S^{+}_{0}+\frac{\bar{z}-\bar{z}_{p}}{{(\Delta\tilde{z}_{p})}}\text{
}S^{+}_{0}\right](\mathbf{x}_{p})$ (144)
$\displaystyle\left(S_{1}^{+}-\frac{t-t_{p}}{2}S^{+}_{0}\right)(\mathbf{x})S^{+}_{1}(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-\frac{i}{2}\left[\frac{(\bar{z}-\bar{z}_{p})^{2}}{{(\Delta\tilde{z}_{p})}^{2}}K+\frac{(\bar{z}-\bar{z}_{p})^{2}}{{(\Delta\tilde{z}_{p})}}\text{
}\partial_{\bar{z}_{p}}S^{+}_{1}+\frac{\bar{z}-\bar{z}_{p}}{{(\Delta\tilde{z}_{p})}}\text{
}2S^{+}_{1}\right](\mathbf{x}_{p})$

Clearly, $S^{+}_{1}$ is not a $1+2$D CC primary field when $K\neq 0$.

We now construct the OPEs involving the $T$ field following similar
procedures. We shall not assume that $T$ is a Lorentz quasi-primary; this fact
will emerge automatically.

Keeping in mind that $S^{+}_{0}$ has $h=\frac{3}{2}$ ,
$\left(P_{1,-1}S^{+}_{0}\right)\sim{{\bm{\xi}}}\cdot
S^{+}_{0}\sim\partial_{t}S^{+}_{0}\sim 0$ and the assumption that $S^{+}_{0}$
is a Lorentz quasi-primary implying $\left(L_{1}S^{+}_{0}\right)=0$ from
(137), from the general form (120) of a $T\Phi$ OPE, we see that the
singularities of the $TS^{+}_{0}$ OPE are completely fixed since we also
assumed that no local field has $h<0$ :

$\displaystyle
T(t,z,\bar{z})S^{+}_{0}(t_{p},z_{p},\bar{z}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{\frac{3}{2}S^{+}_{0}}{(\Delta\tilde{z}_{p})^{2}}+\frac{\partial_{z_{p}}S^{+}_{0}}{\Delta\tilde{z}_{p}}\right](t_{p},z_{p},\bar{z_{p}})$
(145)

Thus, $S^{+}_{0}$ is a $1+2$D CC primary field.

Since $T$ also has integer spin $m=2$ , directly using the bosonic exchange
property and keeping in mind that
$\bar{\partial}^{2}S^{+}_{0}\sim\partial_{t}S^{+}_{0}\sim 0$ and
$\frac{1}{2}\partial\bar{\partial}S^{+}_{0}\sim\partial_{t}T$ from (70), we
readily obtain the following $S^{+}_{0}T$ OPE:

$\displaystyle
iS^{+}_{0}(t,z,\bar{z})T(t_{p},z_{p},\bar{z}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}\left[(\bar{z}-\bar{z}_{p})\left(\frac{\partial_{t_{p}}T}{(\Delta\tilde{z}_{p})}+\frac{\frac{3}{2}\partial_{\bar{z}_{p}}S^{+}_{0}}{{(\Delta\tilde{z}_{p})}^{2}}\right)+\frac{\frac{3}{2}S^{+}_{0}}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{\frac{1}{2}\partial_{z_{p}}S^{+}_{0}}{(\Delta\tilde{z}_{p})}\right](\mathbf{x}_{p})$
(146)

Comparison with (119) reveals the following properties of the $T$ field:

$\displaystyle{{\bm{\xi}}}\cdot T\sim\frac{3}{2}\bar{\partial}S^{+}_{0}\hskip
14.22636pt\text{ ; }\hskip 14.22636pt\bar{{{\bm{\xi}}}}\cdot
T\sim-\frac{1}{2}{\partial}S^{+}_{0}\hskip 14.22636pt\text{ ; }\hskip
14.22636pt\left(P_{0,0}T\right)\sim-\frac{3}{2}S^{+}_{0}$ (147)

The third property ensures that $T$ is not even a $1+2$D CC quasi-primary
field while the first two tell us that the three local fields $\partial
S^{+}_{0}$, $T$ and $\bar{\partial}S^{+}_{0}=P$ transform under the three-
dimensional reducible but indecomposible representation (17) (with
$l=2,a=\frac{3}{2},b=-\frac{1}{2}$) of the spin-boost sub-algebra (15).

The consistent ansatz for the $TT$ OPE is now written following the general
form (120) and remembering that $T$ has $(h,\bar{h})=(2,0)$ (as said before,
we do not assume $T$ to be a Lorentz quasi-primary):

$\displaystyle
T(t,z,\bar{z})T(t_{p},z_{p},\bar{z}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\sum\limits^{2}_{n=1}\frac{\left(L_{n}T\right)}{{(\Delta\tilde{z}_{p})}^{n+2}}+\frac{2T}{(\Delta\tilde{z}_{p})^{2}}+\frac{\partial_{z_{p}}T}{\Delta\tilde{z}_{p}}\right.$
$\displaystyle\hskip
202.01456pt\left.-\frac{t-t_{p}}{2}\left\\{\frac{\partial_{t_{p}}T}{(\Delta\tilde{z}_{p})^{2}}+\frac{3\partial_{\bar{z}_{p}}S^{+}_{0}}{{(\Delta\tilde{z}_{p})}^{3}}\right\\}\right](\mathbf{x}_{p})$
(148)

Using the bosonic exchange property and the fact that the local field
$\left(L_{n}T\right)$ has $(h,\bar{h})=(2-n,0)$ , we find that:

$\displaystyle\left(L_{1}T\right)\sim 0\hskip 14.22636pt;\hskip
14.22636pt\left(L_{2}T\right)\sim\frac{c}{2}$

with $c$ being a constant.

Next, we find that the singular terms in the following ansatz for the
$TS_{1}^{+}$ OPE, obeying (120), is already completely fixed:

$\displaystyle
T(t,z,\bar{z})S^{+}_{1}(t_{p},z_{p},\bar{z}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{S^{+}_{1}}{(\Delta\tilde{z}_{p})^{2}}+\frac{\partial_{z_{p}}S^{+}_{1}}{\Delta\tilde{z}_{p}}-\frac{t-t_{p}}{2}\text{
}\frac{\partial_{t_{p}}S^{+}_{1}}{(\Delta\tilde{z}_{p})^{2}}\right](t_{p},z_{p},\bar{z_{p}})$
(149)

where we have used the facts that $S^{+}_{1}$ has $h=1$ , that it is assumed
to be a Lorentz quasi-primary i.e. $\left(L_{1}S^{+}_{1}\right)\sim 0$ and
that no local field has $h<0$. Using the bosonic exchange property, we
immediately find the $S^{+}_{1}T$ OPE below:

$\displaystyle
S_{1}^{+}(t,z,\bar{z})T(t_{p},z_{p},\bar{z}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-\frac{i}{2}\left[(\bar{z}-\bar{z}_{p})^{2}\left(\frac{\bar{\partial}^{2}_{p}S^{+}_{1}}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{\bar{\partial}_{p}T}{\Delta\tilde{z}_{p}}\right)+(\bar{z}-\bar{z}_{p})\text{
}\frac{2\bar{\partial}_{p}S^{+}_{1}}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{2S^{+}_{1}}{{(\Delta\tilde{z}_{p})}^{2}}\right.$
$\displaystyle\hskip
79.66771pt\left.+(t-t_{p})\left\\{(\bar{z}-\bar{z}_{p})\left(\frac{\frac{3}{2}\bar{\partial}_{p}S^{+}_{0}}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{\partial_{t_{p}}T}{{(\Delta\tilde{z}_{p})}}\right)+\frac{\frac{3}{2}S^{+}_{0}}{{(\Delta\tilde{z}_{p})}^{2}}+\frac{\frac{1}{2}\partial_{p}S^{+}_{0}}{(\Delta\tilde{z}_{p})}\right\\}\right](\mathbf{x}_{p})$
(150)

Comparing with (121), we find that $T$ has $\bar{h}=0$ as expected and
$\left(j^{(+)}_{0}T\right)\sim 0$ which along with $\left(L_{1}T\right)\sim 0$
establishes that $T$ is a Lorentz quasi-primary field.

In the above discussion, we did not find any other local field that mixes with
$S^{+}_{1}$ under Carrollian boost. So, along with the fact that
$\bm{\xi}\cdot S^{+}_{1}\sim 0$ , we conclude that $S^{+}_{1}$ indeed
transforms under a spin-boost irrep.

We also note that in the above OPEs, no local field with $\Delta<0$ appear;
so, the assumption that $\Delta\geq 0$ for all local fields would be
equivalent to the assumption that $h\geq 0$ for all local fields.

The mutual OPEs of the six fields $T_{e}$, $j^{a}_{e}$ and $P_{i}$ are read
off these OPEs to be:

$\displaystyle P_{i}(\mathbf{x})P_{j}(\mathbf{x}_{p})\sim 0$ (151)
$\displaystyle
T_{e}(\mathbf{x})P_{i}(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{\frac{3}{2}P_{i}}{(\Delta\tilde{z}_{p})^{2}}+\frac{\partial_{z_{p}}P_{i}}{\Delta\tilde{z}_{p}}\right](\mathbf{x}_{p})$
(152) $\displaystyle
T_{e}(\mathbf{x})T_{e}(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{\frac{c}{2}}{(\Delta\tilde{z}_{p})^{4}}+\frac{2T_{e}}{(\Delta\tilde{z}_{p})^{2}}+\frac{\partial_{z_{p}}T_{e}}{\Delta\tilde{z}_{p}}\right](\mathbf{x}_{p})$
(153) $\displaystyle
j^{a}_{e}(\mathbf{x})P_{j}(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-\frac{i}{2}\left(\frac{a-1}{2}-j\right)\text{
}\frac{P_{j+a}(\mathbf{x}_{p})}{\Delta\tilde{z}_{p}}$ (154) $\displaystyle
T_{e}(\mathbf{x})j^{a}_{e}(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{j^{a}_{e}}{(\Delta\tilde{z}_{p})^{2}}+\frac{\partial_{z_{p}}j^{a}_{e}}{\Delta\tilde{z}_{p}}\right](\mathbf{x}_{p})$
(155) $\displaystyle
j^{a}_{e}(\mathbf{x})j^{b}_{e}(\mathbf{x}_{p})\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-\frac{i}{2}\left[\frac{Kg^{ab}}{(\Delta\tilde{z}_{p})^{2}}+\frac{(a-b)j^{a+b}_{e}}{\Delta\tilde{z}_{p}}\right](\mathbf{x}_{p})$
(156)

with the non-zero terms of the ‘metric’ $g^{ab}$ being $g^{+-}=g^{-+}=1$ and
$g^{00}=-\frac{1}{2}$.

These OPEs look exactly same as some well-known holomorphic 2D CFT OPEs; e.g.
(153) is the 2D CFT $TT$ OPE giving rise to the Virasoro algebra with central
charge $c$ ; (156) says that in that 2D CFT, $2j^{a}_{e}$ would be the Kac-
Moody currents generating a $\text{sl}(2,\mathbb{R})$ current algebra at level
$K$ ; $P_{i}$ fields would be Kac-Moody primaries of dimension $h=\frac{3}{2}$
, transforming under the two-dimensional representation of
$\text{sl}(2,\mathbb{R})$.

But we are not studying 2D CFT. More precisely, the technology utilized in 2D
CFT is not readily applicable here. So, we should explicitly derive the mode-
algebra from these OPEs. Below we demonstrate one such calculation:

$\displaystyle\left[j^{a}_{n}\text{ },\text{ }P_{k}(\mathbf{x}_{p})\right]$
$\displaystyle=[\frac{2}{2\pi j}\oint\limits_{C_{u}^{\prime}}d\hat{z}\text{
}\hat{z}^{n}\text{ }j^{a}_{e}(t,\hat{z},\hat{\bar{z}})\text{ },\text{
}P_{k}(\mathbf{x}_{p})]\hskip 42.67912pt[\text{definition \eqref{75}}]$
$\displaystyle=\frac{2}{2\pi j}\oint\limits_{C_{u}}d\hat{z}\text{
}\hat{z}^{n}\text{
}\hat{\mathcal{T}}j^{a}_{e}(t_{p}^{+},\hat{z},\hat{\bar{z}})P_{k}(t_{p},\vec{x}_{p})-0\hskip
42.67912pt[\text{contour subtraction: Figure 1}]$
$\displaystyle=\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left(\frac{a-1}{2}-k\right)\frac{1}{2\pi
j}\oint\limits_{C_{u}}d\hat{z}\text{ }\hat{z}^{n}\text{
}\frac{P_{k+a}(\mathbf{x}_{p})}{\hat{z}-z_{p}-j\epsilon 0^{+}}\hskip
42.67912pt[\text{from OPE \eqref{74}}]$
$\displaystyle=-i\left(\frac{a-1}{2}-k\right)z_{p}^{n}P_{k+a}(\mathbf{x}_{p})$

Using now the mode-expansion for $P_{k}$ , given by (129), on both sides and
comparing coefficients of individual powers, one obtains the following
commutator:

$\displaystyle i\left[j^{a}_{n}\text{ },\text{
}P_{m,k}\right]=\left(\frac{a-1}{2}-k\right)P_{m+n,k+a}$

Below we list the complete mode-algebra with $n,m\in\mathbb{Z}$ ;
$j,k\in\\{-1,0\\}$ and $a,b\in\\{0,\pm 1\\}$, derived in a similar way:

$\displaystyle i\left[P_{n,j}\text{ },\text{ }P_{m,k}\right]=0$ (Abelian
super-translations) (157) $\displaystyle i\left[L_{n}\text{ },\text{
}P_{m,j}\right]=\left(\frac{n-1}{2}-m\right)P_{m+n,j}$ (158) $\displaystyle
i\left[L_{n}\text{ },\text{
}L_{m}\right]=\left(n-m\right)L_{m+n}+\frac{c}{12}\left(n^{3}-n\right)\delta_{n+m,0}$
(Holomrphic Virasoro algebra) (159) $\displaystyle i\left[j^{a}_{n}\text{
},\text{ }P_{m,j}\right]=\left(\frac{a-1}{2}-j\right)P_{m+n,j+a}$ (160)
$\displaystyle i\left[L_{n}\text{ },\text{ }j^{a}_{m}\right]=-mj^{a}_{m+n}$
(161) $\displaystyle i\left[j^{a}_{n}\text{ },\text{
}j^{b}_{m}\right]=(a-b)j^{a+b}_{m+n}+2Kg^{ab}n\delta_{n+m,0}$ (
$\overline{\text{sl}(2,\mathbb{R})}$ current algebra) (162)

Thus, the Lorentz i.e. $\text{sl}(2,\mathbb{C})$ symmetry, generated by the
global super-rotation generators, gets infinitely enhanced to a
$\text{Vir}\ltimes\hat{\overline{\text{sl}{(2,\mathbb{R}})}}$ symmetry
governed by the holomorphic super-rotation generators and the generators of
the diffeomorphism (136). Besides, the global Poincare translations get
embedded in the abelian sub-algebra of the $z^{n+1}$ and $z^{n+1}\bar{z}$
super-translation generators. This result, barring the Virasoro central charge
$c$ and the Kac-Moody algebra level $K$, completely agrees with the leading
and the sub-leading conformally soft graviton symmetry algebra in Celestial
holography Banerjee:2020zlg ; Guevara:2021abz ; Banerjee:2021dlm ;
Banerjee:2022wht . As usual, the central charge $c$ and the level $K$ can not
be fixed using the general symmetry principles alone; some model-specific,
dynamical input is necessary to determine them.

Similarly, had $\\{\bar{T},S^{-}_{0},S^{-}_{1}\\}$ been treated as local
fields, the symmetry algebra arising from the (anti-holomorphic sector) OPEs
would be a semi-direct product of the
$\overline{\text{Vir}}\ltimes\hat{{\text{sl}{(2,\mathbb{R}})}}$ algebra and an
abelian algebra of the $\bar{z}^{n+1}$ and $z\bar{z}^{n+1}$ super-translation
generators.

Thus, just as in the Celestial CFT Banerjee:2022wht , the choice of the
generator-fields as the local fields that can appear in the OPEs determines
which symmetry is manifest at the level of the $1+2$D Carrollian conformal
OPEs.

##  6 Conclusions

In this work, we present a study of the general quantum symmetric aspects of
the $1+2$D Carrollian CFT (on flat Carrollian background) by adopting a first-
principle field-theoretic approach. Along the way, we report how these field
theoretic results can be connected to the physics of mass-less scattering in
the $1+3$D bulk AFS and the Celestial holography, thus taking a step towards
the formulation of an AFS/CarrCFT holographic correspondence.

We started by deriving the position-space Ward identities of a source-less
$1+2$D CarrCFT and went on to show how the super-translation and super-
rotation memory effects Strominger:2014pwa ; Pasterski:2015tva , originating
from the gravitational radiation in the $1+3$D bulk AFS, emerge in the $1+2$D
Carrollian CFT upon some simple manipulations of these Ward identities,
manifested by the presence of a temporal step-function Strominger:2017zoo .

We then performed the temporal Fourier transformation (from $t$ to $\omega$)
of the position-space $(t,z,\bar{z})$ Carrollian Ward identities that describe
the memory effects. When the $\omega$ of the generators of those Ward
identities were taken close to zero, the $(n+1)$-point Carrollian correlators
in the $(\omega,z,\bar{z})$ space factorized into a factor of universal form
and the $n$-point Carrollian correlators without the generator. Following the
identification of the $(\omega,z,\bar{z})$ space correlators of the Carrollian
conformal primaries with $\Delta=1$ and transforming under a spin-boost irrep,
with the bulk AFS null-momentum space $S$-matrices Donnay:2022wvx , the
universal factor was recognized to contain both the leading Weinberg:1965nx ;
He:2014laa  and the sub-leading Cachazo:2014fwa ; Kapec:2014opa  soft graviton
theorems of the bulk AFS. Importantly, the pole(s) of this soft-factor at
$\omega=0$ originated from the temporal step-function that bears the imprint
of the memory effects.

The generators of these Ward identities, that were constructed completely out
of the Carrollian EM tensor components but independently of any bulk AFS
metric components, were then interpreted as the leading and the sub-leading
soft graviton fields. Moreover, we also provided the purely Carrollian
construction of the EM tensor of the 2D Celestial CFT Kapec:2016jld ;
Cheung:2016iub . From these constructions, we inferred that the Celestial
(anti-)holomorphic EM tensor and the (positive)negative-helicity
(energetically) sub-leading soft graviton fields are the 2D shadow
transformations of each other, instead of assuming this fact as in
Fotopoulos:2019tpe ; Fotopoulos:2019vac ; Banerjee:2020kaa .

We then moved on to find the quantum conserved charge operators that generate
the extended BMS4 transformations Barnich:2009se ; Barnich:2010ojg ;
Barnich:2010eb ; Barnich:2011mi  on the space of the quantum fields. We could
find the finite charges generating all the super-rotations and the holomorphic
and the anti-holomorphic but the mixed super-translations. All of these finite
quantum charges are different from the expected Carrollian conformal Noether
charges.

The defining property of the temporal step-function allowed us to directly
relate the (covariant) time-ordered OPEs with the corresponding operator
commutation relations via complex contour integrals without any need to go
through a 2D CFT-like radial quantization procedure, similarly as in $1+1$D
CarrCFT Saha:2022gjw . It also led to a $j\epsilon$-form of the Carrollian
conformal Ward identities and the OPEs that helped us completely establish the
definitions of the quantum conserved charges. As opposed to the
$\theta$-prescription, the $j\epsilon$-prescription facilitates a direct use
of the powerful algebraic properties of the OPEs, like associativity and
commutativity.

Meanwhile, following Banerjee:2020zlg ; Polyakov:1987zb , we observed that the
Carrollian conformal Ward identity of the field $S^{+}_{1e}$ (extracted from
$S^{+}_{1}$) which had been identified with the (bulk) positive-helicity sub-
leading energetically soft graviton, can be recast into a form resembling the
holomorphic $\overline{\text{sl}(2,\mathbb{R})}$ Kac-Moody Ward identities, by
decomposing $S^{+}_{1e}$ into the three generating currents each with $h=1$
but different $\bar{h}$. The global transformations corresponding to this
alleged $\overline{\text{sl}(2,\mathbb{R})}$ Kac-Moody symmetry was found to
be the three anti-holomorphic Lorentz transformations.

We now summarize the actions of the (symmetry) generator fields in the $1+2$D
CarrCFT:

  * •

The $S^{+}_{0}$ field, identified with the positive-helicity leading soft
graviton Banerjee:2018fgd ; He:2014laa , embeds two quantum fields $P_{-1}$
and $P_{0}$, both with $h=\frac{3}{2}$. Their modes respectively generate the
$z^{a+1}$ and the $z^{a+1}\bar{z}$ super-translations that include the four
Poincare translations (for $a=-1,0$). The field $S^{-}_{0}$ which is the 2D
shadow transformation of $S^{+}_{0}$, consists of the modes generating the
$\bar{z}^{a+1}$ and the $z\bar{z}^{a+1}$ super-translations.

  * •

The $T$ field consists of two parts: $\partial\bar{\partial}S^{+}_{0}$ and the
field $T_{e}$ which is identified with the holomorphic EM tensor of the
Celestial CFT Kapec:2016jld ; Cheung:2016iub . The modes of $T_{e}$ generate
the holomorphic super-rotations. The modes of the similarly defined field
$\bar{T}_{e}$, embedded into $\bar{T}$, generate the anti-holomorphic super-
rotations as anticipated. The ($\bar{T}_{e}$) $T_{e}$ Ward identity resembles
the (anti-)holomorphic Virasoro one.

  * •

The $S^{\pm}_{1}$ field is identified with the conformally soft Donnay:2018neh
; Puhm:2019zbl  $\pm$ve-helicity sub-leading soft graviton that consists of
$S^{\pm}_{0}$ and $S^{\pm}_{1e}$ identified with the energetically sub-leading
soft graviton. By construction, $S^{\pm}_{1}$ ($S^{\pm}_{1e}$) turned out to
be the 2D shadow transformation of $\bar{T}$ or $T$ ($\bar{T}_{e}$ or
$T_{e}$). $S^{\mp}_{1e}$ embeds three quantum currents ($\bar{j}^{a}_{e}$)
$j^{a}_{e}$ whose Ward identities take the exactly same form as the
(anti-)holomorphic $\text{sl}(2,\mathbb{R})$ Kac-Moody Ward identities.

But, before reading too much into the strong resemblance of the above Ward
identities to the ones appearing in 2D relativistic CFTs, we need to remember
that most of these Carrollian generator fields have $h$ and $\bar{h}$ both
non-zero, unlike in 2D (anti-)holomorphic CFT. So, it would be more
appropriate to extract the quantum symmetry algebra, i.e. the (charge-)algebra
generated by the modes, from the (allowed) mutual OPEs of the six fields
$S^{\pm}_{0}$, $S^{\pm}_{1}$, $T$ and $\bar{T}$. Recalling that all the fields
involved in an OPE must be local and that an OPE is associative, we argued
that among the two sets $\\{T,S^{+}_{1},S^{+}_{0}\\}$ and
$\\{S^{-}_{1},\bar{T},S^{-}_{0}\\}$, only one can be treated as a set of local
fields for the purpose of forming consistent mutual OPEs.

In this work, we chose to treat $T$, $S^{+}_{1}$ and $S^{+}_{0}$ as the local
fields, i.e. we would analyze the holomorphic sector OPEs. Assuming that
$S^{+}_{0}$ and $S^{+}_{1}$ are both Lorentz quasi-primary local fields,
inspired by a similar assumption in Celestial holography Fotopoulos:2019tpe
and that no local field in the holomorphic sector possesses $h<0$, we were
able to completely determine the pole-singularities of the mutual OPEs of $T$,
$S^{+}_{1}$ and $S^{+}_{0}$ following the procedures of Zamolodchikov:1989mz ;
Saha:2022gjw . Namely, we started from the general OPE structures of these
three fields, read off from the corresponding Ward identities, and then
appealed to only the bosonic exchange properties (commutativity) of these
OPEs.

In particular, we did not require any hint from the physics of the bulk AFS
mass-less scattering. Using only the general Carrollian symmetry arguments and
the general algebraic properties of the OPEs was enough to fix the singular
parts of these OPEs. While we did not say anything about the regular parts of
the OPEs, it is easily noted, using the analogy with the usual 2D CFT
Belavin:1984vu  through our discussion in section 4, that they consist of the
descendant local fields whose correlators can be completely determined from
the correlators of the parent primary fields.

Extracting the forms of the mutual OPEs of the six fields $T_{e}$, $j^{a}_{e}$
(or $S^{+}_{1e}$) for $a\in\\{0,\pm 1\\}$ and $P_{i}$ for $i\in\\{-1,0\\}$
from the holomorphic sector OPEs, we could finally conclude using the
$\text{OPE}\longleftrightarrow\text{Commutator}$ prescription developed in
section 4, that the corresponding modes indeed generate a
$\text{Vir}\ltimes\hat{\overline{\text{sl}(2,\mathbb{R})}}$ symmetry algebra
along with an abelian super-translation ideal, perfectly agreeing with the
Celestial holographic conclusions Banerjee:2020zlg ; Guevara:2021abz ;
Banerjee:2021dlm ; Banerjee:2022wht .

Since we traced the $1+2$D CarrCFT roots of only the leading and sub-leading
soft-graviton theorems Weinberg:1965nx ; Cachazo:2014fwa , an obvious next
step is to find out how the whole $w_{1+\infty}$ tower of symmetries
Strominger:2021lvk  can emerge from the Carrollian conformal physics. In
Celestial holography, the $w_{1+\infty}$ symmetry follows completely from the
leading, the sub-leading and the sub-subleading soft graviton theorems
Guevara:2021abz ; Banerjee:2021cly . So, presumably we should first try to
construct a Carrollian conformal field that would correspond to the sub-
subleading soft graviton field. We hope to report on this in a very near
future.

###### Acknowledgements.

The author would like to thank Arjun Bagchi for helpful discussions at various
stages of this work and also for his valuable comments on the initial version
of the draft. A.S. is financially supported by the PMRF fellowship, MHRD,
India.

## References

  * (1) G. ’t Hooft, “Dimensional reduction in quantum gravity,” Conf. Proc. C 930308 (1993), 284-296 [arXiv:gr-qc/9310026 [gr-qc]].
  * (2) L. Susskind, “The World as a hologram,” J. Math. Phys. 36 (1995), 6377-6396 doi:10.1063/1.531249 [arXiv:hep-th/9409089 [hep-th]].
  * (3) J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998), 231-252 doi:10.1023/A:1026654312961 [arXiv:hep-th/9711200 [hep-th]].
  * (4) E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. 2 (1998), 253-291 doi:10.4310/ATMP.1998.v2.n2.a2 [arXiv:hep-th/9802150 [hep-th]].
  * (5) S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B 428 (1998), 105-114 doi:10.1016/S0370-2693(98)00377-3 [arXiv:hep-th/9802109 [hep-th]].
  * (6) S. Pasterski, S. H. Shao and A. Strominger, “Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere,” Phys. Rev. D 96 (2017) no.6, 065026 doi:10.1103/PhysRevD.96.065026 [arXiv:1701.00049 [hep-th]].
  * (7) S. Pasterski and S. H. Shao, “Conformal basis for flat space amplitudes,” Phys. Rev. D 96 (2017) no.6, 065022 doi:10.1103/PhysRevD.96.065022 [arXiv:1705.01027 [hep-th]].
  * (8) S. Banerjee, “Symmetries of free massless particles and soft theorems,” Gen. Rel. Grav. 51 (2019) no.9, 128 doi:10.1007/s10714-019-2609-z [arXiv:1804.06646 [hep-th]].
  * (9) S. Pasterski, “Lectures on celestial amplitudes,” Eur. Phys. J. C 81 (2021) no.12, 1062 doi:10.1140/epjc/s10052-021-09846-7 [arXiv:2108.04801 [hep-th]].
  * (10) A. M. Raclariu, “Lectures on Celestial Holography,” [arXiv:2107.02075 [hep-th]].
  * (11) G. Travaglini, A. Brandhuber, P. Dorey, T. McLoughlin, S. Abreu, Z. Bern, N. E. J. Bjerrum-Bohr, J. Blümlein, R. Britto and J. J. M. Carrasco, et al. “The SAGEX review on scattering amplitudes,” J. Phys. A 55 (2022) no.44, 443001 doi:10.1088/1751-8121/ac8380 [arXiv:2203.13011 [hep-th]].
  * (12) S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. 140 (1965), B516-B524 doi:10.1103/PhysRev.140.B516
  * (13) F. Cachazo and A. Strominger, “Evidence for a New Soft Graviton Theorem,” [arXiv:1404.4091 [hep-th]].
  * (14) A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 07 (2014), 152 doi:10.1007/JHEP07(2014)152 [arXiv:1312.2229 [hep-th]].
  * (15) T. He, V. Lysov, P. Mitra and A. Strominger, “BMS supertranslations and Weinberg’s soft graviton theorem,” JHEP 05 (2015), 151 doi:10.1007/JHEP05(2015)151 [arXiv:1401.7026 [hep-th]].
  * (16) D. Kapec, V. Lysov, S. Pasterski and A. Strominger, “Semiclassical Virasoro symmetry of the quantum gravity $\mathcal{S}$-matrix,” JHEP 08 (2014), 058 doi:10.1007/JHEP08(2014)058 [arXiv:1406.3312 [hep-th]].
  * (17) D. Kapec, P. Mitra, A. M. Raclariu and A. Strominger, “2D Stress Tensor for 4D Gravity,” Phys. Rev. Lett. 119 (2017) no.12, 121601 doi:10.1103/PhysRevLett.119.121601 [arXiv:1609.00282 [hep-th]].
  * (18) C. Cheung, A. de la Fuente and R. Sundrum, “4D scattering amplitudes and asymptotic symmetries from 2D CFT,” JHEP 01 (2017), 112 doi:10.1007/JHEP01(2017)112 [arXiv:1609.00732 [hep-th]].
  * (19) T. He, D. Kapec, A. M. Raclariu and A. Strominger, “Loop-Corrected Virasoro Symmetry of 4D Quantum Gravity,” JHEP 08 (2017), 050 doi:10.1007/JHEP08(2017)050 [arXiv:1701.00496 [hep-th]].
  * (20) L. Donnay, K. Nguyen and R. Ruzziconi, “Loop-corrected subleading soft theorem and the celestial stress tensor,” JHEP 09 (2022), 063 doi:10.1007/JHEP09(2022)063 [arXiv:2205.11477 [hep-th]].
  * (21) H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, “Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems,” Proc. Roy. Soc. Lond. A 269 (1962), 21-52 doi:10.1098/rspa.1962.0161
  * (22) R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times,” Proc. Roy. Soc. Lond. A 270 (1962), 103-126 doi:10.1098/rspa.1962.0206
  * (23) R. Sachs, “Asymptotic symmetries in gravitational theory,” Phys. Rev. 128 (1962), 2851-2864 doi:10.1103/PhysRev.128.2851
  * (24) G. Barnich and C. Troessaert, “Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited,” Phys. Rev. Lett. 105 (2010), 111103 doi:10.1103/PhysRevLett.105.111103 [arXiv:0909.2617 [gr-qc]].
  * (25) G. Barnich and C. Troessaert, “Supertranslations call for superrotations,” PoS CNCFG2010 (2010), 010 doi:10.22323/1.127.0010 [arXiv:1102.4632 [gr-qc]].
  * (26) G. Barnich and C. Troessaert, “Aspects of the BMS/CFT correspondence,” JHEP 05 (2010), 062 doi:10.1007/JHEP05(2010)062 [arXiv:1001.1541 [hep-th]].
  * (27) G. Barnich and C. Troessaert, “BMS charge algebra,” JHEP 12 (2011), 105 doi:10.1007/JHEP12(2011)105 [arXiv:1106.0213 [hep-th]].
  * (28) A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, “Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory,” Nucl. Phys. B 241 (1984), 333-380 doi:10.1016/0550-3213(84)90052-X
  * (29) A. B. Zamolodchikov, “Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory,” Theor. Math. Phys. 65 (1985), 1205-1213 doi:10.1007/BF01036128
  * (30) A. B. Zamolodchikov, A. B. Zamolodchikov and I. M. Khalatnikov, “PHYSICS REVIEWS. VOL. 10, PT. 4: CONFORMAL FIELD THEORY AND CRITICAL PHENOMENA IN TWO-DIMENSIONAL SYSTEMS,”
  * (31) J. Distler, R. Flauger and B. Horn, “Double-soft graviton amplitudes and the extended BMS charge algebra,” JHEP 08 (2019), 021 doi:10.1007/JHEP08(2019)021 [arXiv:1808.09965 [hep-th]].
  * (32) W. Fan, A. Fotopoulos and T. R. Taylor, “Soft Limits of Yang-Mills Amplitudes and Conformal Correlators,” JHEP 05 (2019), 121 doi:10.1007/JHEP05(2019)121 [arXiv:1903.01676 [hep-th]].
  * (33) M. Pate, A. M. Raclariu and A. Strominger, “Conformally Soft Theorem in Gauge Theory,” Phys. Rev. D 100 (2019) no.8, 085017 doi:10.1103/PhysRevD.100.085017 [arXiv:1904.10831 [hep-th]].
  * (34) A. Guevara, “Notes on Conformal Soft Theorems and Recursion Relations in Gravity,” [arXiv:1906.07810 [hep-th]].
  * (35) A. Fotopoulos and T. R. Taylor, “Primary Fields in Celestial CFT,” JHEP 10 (2019), 167 doi:10.1007/JHEP10(2019)167 [arXiv:1906.10149 [hep-th]].
  * (36) M. Pate, A. M. Raclariu, A. Strominger and E. Y. Yuan, “Celestial operator products of gluons and gravitons,” Rev. Math. Phys. 33 (2021) no.09, 2140003 doi:10.1142/S0129055X21400031 [arXiv:1910.07424 [hep-th]].
  * (37) A. Fotopoulos, S. Stieberger, T. R. Taylor and B. Zhu, “Extended BMS Algebra of Celestial CFT,” JHEP 03 (2020), 130 doi:10.1007/JHEP03(2020)130 [arXiv:1912.10973 [hep-th]].
  * (38) S. Banerjee, S. Ghosh and R. Gonzo, “BMS symmetry of celestial OPE,” JHEP 04 (2020), 130 doi:10.1007/JHEP04(2020)130 [arXiv:2002.00975 [hep-th]].
  * (39) S. Banerjee, S. Ghosh and P. Paul, “MHV graviton scattering amplitudes and current algebra on the celestial sphere,” JHEP 02 (2021), 176 doi:10.1007/JHEP02(2021)176 [arXiv:2008.04330 [hep-th]].
  * (40) S. Banerjee and S. Ghosh, “MHV gluon scattering amplitudes from celestial current algebras,” JHEP 10 (2021), 111 doi:10.1007/JHEP10(2021)111 [arXiv:2011.00017 [hep-th]].
  * (41) A. Guevara, E. Himwich, M. Pate and A. Strominger, “Holographic symmetry algebras for gauge theory and gravity,” JHEP 11 (2021), 152 doi:10.1007/JHEP11(2021)152 [arXiv:2103.03961 [hep-th]].
  * (42) S. Banerjee, S. Ghosh and S. S. Samal, “Subsubleading soft graviton symmetry and MHV graviton scattering amplitudes,” JHEP 08 (2021), 067 doi:10.1007/JHEP08(2021)067 [arXiv:2104.02546 [hep-th]].
  * (43) A. Strominger, “w(1+infinity) and the Celestial Sphere,” [arXiv:2105.14346 [hep-th]].
  * (44) S. Banerjee, S. Ghosh and P. Paul, “(Chiral) Virasoro invariance of the tree-level MHV graviton scattering amplitudes,” JHEP 09 (2022), 236 doi:10.1007/JHEP09(2022)236 [arXiv:2108.04262 [hep-th]].
  * (45) C. Duval, G. W. Gibbons and P. A. Horvathy, “Conformal Carroll groups and BMS symmetry,” Class. Quant. Grav. 31 (2014), 092001 doi:10.1088/0264-9381/31/9/092001 [arXiv:1402.5894 [gr-qc]].
  * (46) C. Duval, G. W. Gibbons and P. A. Horvathy, “Conformal Carroll groups,” J. Phys. A 47 (2014) no.33, 335204 doi:10.1088/1751-8113/47/33/335204 [arXiv:1403.4213 [hep-th]].
  * (47) C. Duval, G. W. Gibbons, P. A. Horvathy and P. M. Zhang, “Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time,” Class. Quant. Grav. 31 (2014), 085016 doi:10.1088/0264-9381/31/8/085016 [arXiv:1402.0657 [gr-qc]].
  * (48) J. M. Lévy-Leblond, “Une nouvelle limite non-relativiste du group de Poincaré,” Ann. Inst. H Poincaré 3 (1965) 1
  * (49) N. D. Sen Gupta, “On an Analogue of the Galileo Group,” Il Nuovo Cimento 44 (1966) 512
  * (50) G. Barnich, A. Gomberoff and H. A. Gonzalez, “The Flat limit of three dimensional asymptotically anti-de Sitter spacetimes,” Phys. Rev. D 86 (2012), 024020 doi:10.1103/PhysRevD.86.024020 [arXiv:1204.3288 [gr-qc]].
  * (51) G. Barnich, “Entropy of three-dimensional asymptotically flat cosmological solutions,” JHEP 10 (2012), 095 doi:10.1007/JHEP10(2012)095 [arXiv:1208.4371 [hep-th]].
  * (52) A. Bagchi, S. Detournay, R. Fareghbal and J. Simón, “Holography of 3D Flat Cosmological Horizons,” Phys. Rev. Lett. 110 (2013) no.14, 141302 doi:10.1103/PhysRevLett.110.141302 [arXiv:1208.4372 [hep-th]].
  * (53) G. Barnich and H. A. Gonzalez, “Dual dynamics of three dimensional asymptotically flat Einstein gravity at null infinity,” JHEP 05 (2013), 016 doi:10.1007/JHEP05(2013)016 [arXiv:1303.1075 [hep-th]].
  * (54) A. Bagchi, R. Basu, D. Grumiller and M. Riegler, “Entanglement entropy in Galilean conformal field theories and flat holography,” Phys. Rev. Lett. 114 (2015) no.11, 111602 doi:10.1103/PhysRevLett.114.111602 [arXiv:1410.4089 [hep-th]].
  * (55) H. Jiang, W. Song and Q. Wen, “Entanglement Entropy in Flat Holography,” JHEP 07 (2017), 142 doi:10.1007/JHEP07(2017)142 [arXiv:1706.07552 [hep-th]].
  * (56) E. Hijano and C. Rabideau, “Holographic entanglement and Poincaré blocks in three-dimensional flat space,” JHEP 05 (2018), 068 doi:10.1007/JHEP05(2018)068 [arXiv:1712.07131 [hep-th]].
  * (57) A. Bagchi, D. Grumiller and W. Merbis, “Stress tensor correlators in three-dimensional gravity,” Phys. Rev. D 93 (2016) no.6, 061502 doi:10.1103/PhysRevD.93.061502 [arXiv:1507.05620 [hep-th]].
  * (58) J. Hartong, “Holographic Reconstruction of 3D Flat Space-Time,” JHEP 10 (2016), 104 doi:10.1007/JHEP10(2016)104 [arXiv:1511.01387 [hep-th]].
  * (59) A. Bagchi, R. Basu, A. Kakkar and A. Mehra, “Flat Holography: Aspects of the dual field theory,” JHEP 12 (2016), 147 doi:10.1007/JHEP12(2016)147 [arXiv:1609.06203 [hep-th]].
  * (60) L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, “Carrollian Perspective on Celestial Holography,” Phys. Rev. Lett. 129 (2022) no.7, 071602 doi:10.1103/PhysRevLett.129.071602 [arXiv:2202.04702 [hep-th]].
  * (61) A. Bagchi, S. Banerjee, R. Basu and S. Dutta, “Scattering Amplitudes: Celestial and Carrollian,” Phys. Rev. Lett. 128 (2022) no.24, 241601 doi:10.1103/PhysRevLett.128.241601 [arXiv:2202.08438 [hep-th]].
  * (62) L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, “Bridging Carrollian and Celestial Holography,” [arXiv:2212.12553 [hep-th]].
  * (63) A. Strominger and A. Zhiboedov, “Gravitational Memory, BMS Supertranslations and Soft Theorems,” JHEP 01 (2016), 086 doi:10.1007/JHEP01(2016)086 [arXiv:1411.5745 [hep-th]].
  * (64) S. Pasterski, A. Strominger and A. Zhiboedov, “New Gravitational Memories,” JHEP 12 (2016), 053 doi:10.1007/JHEP12(2016)053 [arXiv:1502.06120 [hep-th]].
  * (65) A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,” [arXiv:1703.05448 [hep-th]].
  * (66) A. Laddha and A. Sen, “Sub-subleading Soft Graviton Theorem in Generic Theories of Quantum Gravity,” JHEP 10 (2017), 065 doi:10.1007/JHEP10(2017)065 [arXiv:1706.00759 [hep-th]].
  * (67) L. Freidel, D. Pranzetti and A. M. Raclariu, “Sub-subleading soft graviton theorem from asymptotic Einstein’s equations,” JHEP 05 (2022), 186 doi:10.1007/JHEP05(2022)186 [arXiv:2111.15607 [hep-th]].
  * (68) L. P. de Gioia and A. M. Raclariu, “Celestial Sector in CFT: Conformally Soft Symmetries,” [arXiv:2303.10037 [hep-th]].
  * (69) S. Banerjee, “Null Infinity and Unitary Representation of The Poincare Group,” JHEP 01 (2019), 205 doi:10.1007/JHEP01(2019)205 [arXiv:1801.10171 [hep-th]].
  * (70) A. Saha, “Intrinsic approach to 1 + 1D Carrollian Conformal Field Theory,” JHEP 12 (2022), 133 doi:10.1007/JHEP12(2022)133 [arXiv:2207.11684 [hep-th]].
  * (71) P. Di Francesco, P. Mathieu and D. Senechal, “Conformal Field Theory,” Springer-Verlag, 1997, ISBN 978-0-387-94785-3, 978-1-4612-7475-9
  * (72) A. M. Polyakov, “Quantum Gravity in Two-Dimensions,” Mod. Phys. Lett. A 2 (1987), 893 doi:10.1142/S0217732387001130
  * (73) S. Banerjee and S. Pasterski, “Revisiting the Shadow Stress Tensor in Celestial CFT,” [arXiv:2212.00257 [hep-th]].
  * (74) T. Klose, T. McLoughlin, D. Nandan, J. Plefka and G. Travaglini, “Double-Soft Limits of Gluons and Gravitons,” JHEP 07 (2015), 135 doi:10.1007/JHEP07(2015)135 [arXiv:1504.05558 [hep-th]].
  * (75) A. Bagchi, R. Chatterjee, R. Kaushik, A. Saha and D. Sarkar, “Non-Lorentzian Kač-Moody algebras,” JHEP 03 (2023), 041 doi:10.1007/JHEP03(2023)041 [arXiv:2301.04686 [hep-th]].
  * (76) M. Campiglia and A. Laddha, “Asymptotic symmetries and subleading soft graviton theorem,” Phys. Rev. D 90 (2014) no.12, 124028 doi:10.1103/PhysRevD.90.124028 [arXiv:1408.2228 [hep-th]].
  * (77) M. Campiglia and A. Laddha, “New symmetries for the Gravitational S-matrix,” JHEP 04 (2015), 076 doi:10.1007/JHEP04(2015)076 [arXiv:1502.02318 [hep-th]].
  * (78) G. Compère, A. Fiorucci and R. Ruzziconi, “Superboost transitions, refraction memory and super-Lorentz charge algebra,” JHEP 11 (2018), 200 [erratum: JHEP 04 (2020), 172] doi:10.1007/JHEP11(2018)200 [arXiv:1810.00377 [hep-th]].
  * (79) L. Donnay, S. Pasterski and A. Puhm, “Asymptotic Symmetries and Celestial CFT,” JHEP 09 (2020), 176 doi:10.1007/JHEP09(2020)176 [arXiv:2005.08990 [hep-th]].
  * (80) M. Campiglia and J. Peraza, “Generalized BMS charge algebra,” Phys. Rev. D 101 (2020) no.10, 104039 doi:10.1103/PhysRevD.101.104039 [arXiv:2002.06691 [gr-qc]].
  * (81) B. Oblak, “From the Lorentz Group to the Celestial Sphere,” [arXiv:1508.00920 [math-ph]].
  * (82) R. Blumenhagen and E. Plauschinn, “Introduction to conformal field theory: with applications to String theory,” Lect. Notes Phys. 779 (2009), 1-256
  * (83) B. Chen, R. Liu and Y. f. Zheng, “On Higher-dimensional Carrollian and Galilean Conformal Field Theories,” [arXiv:2112.10514 [hep-th]].
  * (84) A. Bagchi, R. Basu, A. Mehra and P. Nandi, “Field Theories on Null Manifolds,” JHEP 02 (2020), 141 doi:10.1007/JHEP02(2020)141 [arXiv:1912.09388 [hep-th]].
  * (85) S. Baiguera, G. Oling, W. Sybesma and B. T. Søgaard, “Conformal Carroll Scalars with Boosts,” [arXiv:2207.03468 [hep-th]].
  * (86) C. Itzykson and J. B. Zuber, “Quantum Field Theory,” McGraw-Hill, 1980, ISBN 978-0-486-44568-7
  * (87) B. Chen, R. Liu, H. Sun and Y. f. Zheng, “Constructing Carrollian Field Theories from Null Reduction,” [arXiv:2301.06011 [hep-th]].
  * (88) M. J. Ablowitz, A. S. Fokas, “Complex Variables: Introduction and Applications,” Second edition, Cambridge University Press, 2003.
  * (89) S. Banerjee, P. Pandey and P. Paul, “Conformal properties of soft operators: Use of null states,” Phys. Rev. D 101 (2020) no.10, 106014 doi:10.1103/PhysRevD.101.106014 [arXiv:1902.02309 [hep-th]].
  * (90) S. Banerjee and P. Pandey, “Conformal properties of soft-operators. Part II. Use of null-states,” JHEP 02 (2020), 067 doi:10.1007/JHEP02(2020)067 [arXiv:1906.01650 [hep-th]].
  * (91) L. Donnay, A. Puhm and A. Strominger, “Conformally Soft Photons and Gravitons,” JHEP 01 (2019), 184 doi:10.1007/JHEP01(2019)184 [arXiv:1810.05219 [hep-th]].
  * (92) A. Puhm, “Conformally Soft Theorem in Gravity,” JHEP 09 (2020), 130 doi:10.1007/JHEP09(2020)130 [arXiv:1905.09799 [hep-th]].
  * (93) X. Bekaert and B. Oblak, “Massless scalars and higher-spin BMS in any dimension,” JHEP 11 (2022), 022 doi:10.1007/JHEP11(2022)022 [arXiv:2209.02253 [hep-th]].
  * (94) W. B. Liu and J. Long, “Symmetry group at future null infinity I: scalar theory,” [arXiv:2210.00516 [hep-th]].
  * (95) G. Barnich, “Centrally extended BMS4 Lie algebroid,” JHEP 06 (2017), 007 doi:10.1007/JHEP06(2017)007 [arXiv:1703.08704 [hep-th]].
  * (96) L. Donnay and R. Ruzziconi, “BMS flux algebra in celestial holography,” JHEP 11 (2021), 040 doi:10.1007/JHEP11(2021)040 [arXiv:2108.11969 [hep-th]].
  * (97) J. H. Schwarz, “Diffeomorphism Symmetry in Two Dimensions and Celestial Holography,” [arXiv:2208.13304 [hep-th]].