The analytic structure of the inversion function of MFT for $\xi_2>\xi_1>0$. There are five branch points at $\xi =\pm \xi_1 \pm \xi_2,0$ drawn as the wavy lines. The physical cut corresponding to the double-trace operators is anchored at $\xi=\xi_1+\xi_2$. The dashed lines are different analytic continuations to the imaginary axis, and from this ambiguity we read off the discontinuity along the cut $\xi=\xi_1+\xi_2$.

Dispersion relation. Notice that $c^{\text{u.p.s.}}(\D,\xi)$ has no poles of $(\D,\xi)$, and instead there are five branch cuts anchored at $\xi =\pm \xi_1 \pm \xi_2,0$ shown in figure <ref>. This phenomenon is expected as explained in <cit.>. In the case of four identical external operators, the physical inversion function $c(\D,\xi)$ has two branch cuts, and $c^{\text{u.p.s.}}(\D,\xi)$ with respect to virtual external operators is the imaginary part of $c(\D,\xi)$ along the physical cut, due to the ambiguity of analytic continuation of weights. The prototypical example of this ambiguity is

\begin{equation}
\intt{dt}\frac{f(t)}{t\pm i\e}=\PV\intt{dt} \frac{f(t)}{t} \mp i\pi f(0).
\end{equation}

The inversion function $c^{\text{u.p.s.}}(\D,\xi)$ plays the role of $-\pi f(0)$ and $c(\D,\xi)$ is multi-valued. For the preceding reason we propose the dispersion relation

\begin{equation}
\disc_{\text{phy. cut}} c(\D,\xi)= \lim_{\e\to 0}\bpair{c(\D,\xi+\e)-c(\D,\xi-\e)}
2i\,  c^{\text{u.p.s.}}(\D,\xi),
\label{eq:dispersion}
\end{equation}

and then

\begin{equation}
\disc_{\text{phy. cut}} I(\D,\xi)
2i\,  I^{\text{u.p.s.}}(\D,\xi).
\end{equation}

The real part is recovered by the Kramers–Kronig relation[It is also known as the Hilbert transform, and will be reviewed in appendix <ref>.],

\begin{equation}
\Re c(\D,\xi)=\frac{1}{\pi}\PV\inttt{d\xi'}{\R} \frac{1}{\xi'-\xi}c^{\text{u.p.s.}}(\D,\xi').
\end{equation}

A natural question is that whether the inversion function can be continued onto the unitary principal series. This requires the external operators of the input four-point functions can be continued according to the scheme $\xi_i\to e^{i\th}\xi_i$ shown in figure <ref>. In other words we need a family of four-point functions $G_4(\lm)$ depending on $\lm$ analytically. In relativistic CFT this is rare since the unitary theories are expected to be isolated points in the space of CFTs. In GCFT$_2$ we show in the appendix <ref> that the family $G_4(\lm)$ exists at least for real $\lm$, due to the existence of outer-automorphism of the Galilean conformal algebra. We leave this question for further study and return to the discussion on MFT.

Inversion formula. After inserting the inversion function into the inversion formula (<ref>), we enclose the $\xi$-contour along the physical branch cut,

\begin{equation}
\vev{\f_1\f_2\f_2\f_1}_{\cP}
\int^{1+i\inf}_{1-i\inf}\frac{d\D}{2\pi i}
\int^{+\inf}_{\xi_1+\xi_2}\frac{d\xi}{2\pi i}\disc c(\D,\xi)\blockdressed_{\D,\xi},
\end{equation}

then using the relation (<ref>) we get

\begin{align}
\rhs
&=\int^{1+i\inf}_{1-i\inf}\frac{d\D}{2\pi i}
\int^{+\inf}_{\xi_1+\xi_2}\,\frac{d\xi}{\pi}\, c^{\text{u.p.s.}}(\D,\xi)\blockdressed_{\D,\xi}\nn
\\
\int^{1+i\inf}_{1-i\inf}\frac{d\D}{2\pi i}
\int^{+\inf}_{\xi_1+\xi_2}\,d\xi\,\left(\xi-\xi_{1}-\xi_{2}\right)^{-1-\D+\D_{1}+\D_{2}}F(\xi,\xi_1,\xi_2)\blockdressed_{\D,\xi}.
\label{eq:inversion11}
\end{align}

As discussed in the appendix <ref>, the power factor $\left(\xi-\xi_{1}-\xi_{2}\right)^{-1-\D+\D_{1}+\D_{2}}$ as tempered distribution possesses simple poles at

\begin{equation}
\D_n=\D_{1}+\D_{2}+n, \hspace{3ex} n=0,1,2,\dots .
\label{eq:doubletracepoles}
\end{equation}

The remaining factors are analytic at $\xi=\xi_1+\xi_2$ and the Taylor expansion is

\begin{equation}
F(\xi,\xi_1,\xi_2)\blockdressed_{\D,\xi}=\sum_{n=0}^{\inf} a_n(\D)\left(\xi-\xi_{1}-\xi_{2}\right)^n.
\end{equation}

Then separating the $\xi$-integration of (<ref>) into two parts[Choosing different separations does not affect the analytic structure and give the same result. The contributions come from the singularity $\xi=\xi_1+\xi_2$.], we find that the part on the interval $(\xi_1+\xi_2+1,\inf)$ contributes no $\D$-poles and hence can be dropped when deforming the $\D$-contour,

\begin{align}
\rhs
% &=
% \int^{1+i\inf}_{1-i\inf}\frac{d\D}{2\pi i}
% \int^{+\inf}_{\xi_1+\xi_2}\sum_{n=0}^{\inf} a_n\,d\xi\,\left(\xi-\xi_{1}-\xi_{2}\right)^{-1-\D+\D_{1}+\D_{2}+n}
% \\
\int^{1+i\inf}_{1-i\inf}\frac{d\D}{2\pi i}
\bpair{
\ppair{\int^{+\inf}_{\xi_1+\xi_2+1}d\xi + \int^{\xi_1+\xi_2+1}_{\xi_1+\xi_2}d\xi}
\sum_{n=0}^{\inf} \left(\xi-\xi_{1}-\xi_{2}\right)^{-1-\D+\D_{1}+\D_{2}+n}
\nn
\\
\int^{1+i\inf}_{1-i\inf}\frac{d\D}{2\pi i}
\sum_{n=0}^{\inf}
\int^{\xi_1+\xi_2+1}_{\xi_1+\xi_2}d\xi \left(\xi-\xi_{1}-\xi_{2}\right)^{-1-\D+\D_{1}+\D_{2}+n}a_n(\D)
\nn
\\
-\int^{1+i\inf}_{1-i \inf}\frac{d\D}{2\pi i}
\sum_{n=0}^{\inf}\frac{a_n(\D)}{\D-\D_{1}-\D_{2}-n}
\nn
\\
\sum_{n=0}^{\inf}a_n(\D_{1}+\D_{2}+n),
\label{eq:doubletrace1}
\end{align}

where in the last line we have deformed the $\D$-contour and pick up the double-trace poles (<ref>), and the minus sign is canceled due to the clock-wise order of the contour.

Each term in the summation (<ref>) fits into a conformal block of rank-$(n+1)$ boost multiplet with double-trace weight $(\D,\xi)=(\D_1+\D_2+n,\xi_1+\xi_1)$

\begin{equation}
\end{equation}

where the block coefficients are

\begin{equation}
\label{eq:block}
\end{equation}

and they match with (<ref>).

In summary we have the conformal block expansion of the projected four-point function

\begin{equation}
\label{eq:projectedfourpoint}
\vev{\f_1\f_2\f_2\f_1}_{\cP}
\end{equation}

with the block coefficients (<ref>).
In the crossing region $(0,1)\times\RR$, the projected four-point function (<ref>) converges rapidly and can be re-expanded as a double Taylor series of $(x,k)$ in the limit. The expansion matches with that of the four-point function (<ref>) order by order. Hence (<ref>) should equal to (<ref>) in the crossing region. In this way we derive the conformal block expansion of the four-point function and confirm the validity of the inversion formula.

§.§ Bilocal actions of mean field theory

There are few GCFT$_2$ models with concrete actions. One of them that has been thoroughly discussed in
the literature is the BMS free scalar model <cit.> with the action on the plane

\begin{equation}
S=\half \intt{dxdy}\f \p_{y}^2 \f.
\label{eq:actionBMSfreescalar}
\end{equation}

As a worldsheet theory, this model describes the tensionless limit of the free bosonic string, see e.g. <cit.> and the ambitwistor strings <cit.>. This action can also be realized as a $\sqrt{T\bar{T}}$ deformation of the relativistic free scalar <cit.>. The higher dimensional Carrollian analog of (<ref>) was discussed in <cit.>.

In this subsection we explore the Lagrangian description of Galilean MFT. Inspired by the MFT in AdS/CFT <cit.> and the long-range models in statistical physics, see e.g. <cit.>, we find a series of bilocal actions labeled by $(\D,\xi)$ corresponding to the Galilean MFT. Moreover at the special value $\xi=0$ we get additional actions labeled by $(\D_1,\D_2)$, one of which gives the BMS free scalar (<ref>). Starting from the ansatz of bilocal action

\begin{equation}
\label{eq:actionbilocal1}
\end{equation}

with two scalars transforming as $\f_i\in \rep{\D_i,\xi_i}$, and imposing the Galilean conformal invariance as in table <ref> on the action, we get four equations of $K$, which are related to the two-point Ward identities (<ref>) by the shadow replacement $(\D_1,\xi_1,\D_2,\xi_2)\to(2-\D_1,-\xi_1,2-\D_2,-\xi_2)$.

Mathematically speaking, the discussion in section <ref> is on the intertwining maps, and the action (<ref>) is an intertwining bilinear form introduced by Bruhat <cit.>, see also chapter 3 of <cit.>. Intuitively, the finite dimensional analogs for two representations $\cV_i\ni V_i$ are

\begin{equation}
V_{1}^{a}=K^{a}_{b}V_{2}^{b},\qquad (V_1,V_2)_{K}=K_{ab}V_{1}^{a}V_{2}^{b}.
\end{equation}

For unitary representations the two concepts are essentially the same. We can use the positive-definite inner product $(V_1,U_1)=g_{ab} V_{1}^{a} U_{1}^{b}$ to raise and lower the indices, then the intertwining map and bilinear form are related by $K_{ac}=g_{ab} K^{b}_{c}$. For non-unitary representations they are not necessarily equivalent.

Bilocal actions. By solving the differential equations of $K$, we can read the bilocal actions. We find the following distributional solutions for $K$ in two different cases.

Case 1: $\xi_1+\xi_2\neq 0$. The last two equations in (<ref>) force the scalars to be identical, $\xi_1=\xi_2,\,\D_1=\D_2$, and the solutions are exactly the kernel of the shadow transform (<ref>),

\begin{equation}
K(x_{12},y_{12})=e^{-2\xi_1\frac{y_{12}}{x_{12}}}\bpair{c_1|x_{12}|^{2\D_1-4}+c_2 \sign(x_{12})|x_{12}|^{2\D_1-4}},
\label{eq:actiontypical}
\end{equation}

where the first one corresponds to bosonic statistics and the second to fermionic statistics. At special values $\xi_1=0,\, \D_1=\frac{3-n}{2},\,n\in \Zgeq$, one of the power-type distributions should be regularized to the $n$-th derivative of the $\d$-distribution $^̣(n)(x_12)$, and the action is local with respect to $x$.

Case 2: $ξ_1+ξ_2= 0$. Besides the previous power-law solutions, the first two equations in \eqref{eq:wardsinglet} admit an additional solution
\begin{equation}
K(x_{12},y_{12})= \d(x_{12})\bpair{d_1|y_{12}|^{\D_1+\D_2-3}+d_2 \sign(y_{12})|y_{12}|^{\D_1+\D_2-3}}.
\label{eq:actionanomalous}
\end{equation}
The third equation gives no constraint and the last equation sets $ξ_1=ξ_2=0$. Notice that in this case the scalars are not compulsory to be equal. At special values $_1+_2=2-n, n∈$, one of the power-type distributions in the additional solutions \eqref{eq:actionanomalous} should be regularized to the $n$-th derivative of the $$̣-distribution $\d^{(n)}(y_{12})$, and the action turns out to be local and BMS-invariant,

\begin{equation}
\end{equation}

For example, in the case $\xi_1=\xi_2=0,\, \D_1+\D_2=0$ there are totally four actions as follows,

\begin{align}
S& =\intt{dx_1dy_1dx_2dy_2}\f(\xy{1})\f(\xy{2})x_{12}^{-4},
\label{eq:actionbad1}
\\
S& =\intt{dxdy_1dy_2}\f(x,y_1)\p_{x}^{3}\f(x,y_2),
\label{eq:actionbad2}
\\
S& =\intt{dxdy_1dy_2}\f_1(x,y_1)\f_2(x,y_2)y_{12}^{-3},
\\
S& =\intt{dxdy}\f_1(x,y)\p_{y}^{2}\f_2(x,y),
\end{align}

and when the two fields are identical, the last one is the BMS free scalar model (<ref>). Another example is that if $\f_1=\f_2=\psi,\, \D_1=\half,\, \xi_1=\xi_2=0$ and setting $\psi$ to be Grassmann-valued, we get the fermionic part of the homogeneous tensionless superstring <cit.>,

\begin{equation}
\end{equation}

Path integral quantization. These actions (<ref>) and (<ref>) are free and the correlation functions can be evaluated from the path integral. Assuming the two scalars are identically bosonic and normalizing the action by a factor $\half$, the partition function with the source is

\begin{equation}
\end{equation}

Substituting $\f(\xy{1})$ with $ \f(\xy{1})+\intt{dx_2dy_2}K^{-1}(x_{12},y_{12})J(\xy{2})$, where $K^{-1}$ is the inverse of $K$, the partition function is solved as

\begin{equation}
Z[J]=Z[0]\intt{D\f}e^{-\half \intt{dx_1dy_1dx_2dy_2}J(\xy{1})K^{-1}(x_{12},y_{12})J(\xy{2})},
\end{equation}

then the two-point function is $\vev{\f\f}=-K^{-1}$ and the higher-point functions are the Wick contractions of $\vev{\f\f}$.

For the first series of actions (<ref>) and $\xi\in\Rnot$, we need to start from the imaginary Carrollian or Galilean time at the beginning as in section <ref>, and the integral equation of the inverse kernel is

\begin{equation}
\int dx_0dy_0\,e^{-2\xi\frac{y_{10}}{x_{10}}}|x_{10}|^{2\D-4}K^{-1}(x_{02},y_{02})=\d(x_{12},y_{12}).
\end{equation}

Using (<ref>) the two-point function is

\begin{equation}
\vev{\f(\xy{1})\f(\xy{2})}=c\, e^{2\xi\frac{y_{12}}{x_{12}}}|x_{12}|^{-2\D},
\end{equation}

where $c$ is an unimportant constant, and we get back to the scalar MFT in section <ref>.

Notice that when $\xi=0$ the first series of actions (<ref>),

\begin{equation}
S=\intt{dx_1dx_2 dy_1dy_2} \f(\xy{1})|x_{12}|^{2\D-4}\f(\xy{2}),
\end{equation}

possess gauge redundancies. Any field configuration $\f(x,y)$ satisfying

\begin{equation}
\intt{dy}\f(x,y)=:\Phi(x),
\end{equation}

gives the same contribution $e^{-\intt{dx_1dx_2}\Phi(x_1)|x_{12}|^{2\D-4}\Phi(x_2)}$ to the path integral. In other words, the bilocal kernel $K(x_{12},y_{12})=|x_{12}|^{2\D-4}$ is highly degenerate and has no inverse unless we first mod out the zero-modes, and the two-point function should satisfy

\begin{equation}
\intt{dx_0}|x_{10}|^{2\D-4}\vev{\Phi(x_0)\Phi(x_2)}=-\d(x_{12}),
\end{equation}

instead of the naive one,

\begin{equation}
\intt{dx_0 dy_0}|x_{10}|^{2\D-4}\vev{\f(\xy{0})\f(\xy{2})}=-\d(x_{12})\d(y_{12}).
\end{equation}

Therefore the theory is equivalent to the MFT in CFT$_1$. The action is

\begin{equation}
\end{equation}

with $\D_{\Phi}=\D-1$ and the two-point function is

\begin{equation}
\vev{\Phi(x_1)\Phi(x_2)}=c\, |x_{12}|^{2-2\D}.
\end{equation}

The second series of actions (<ref>) are local with respect to $x$,

\begin{equation}
S=\intt{dx dy_1dy_2} \f(x,y_1)|y_{12}|^{2\D-3}\f(x,y_2).
\end{equation}

The two-point function satisfies the following integral equation,

\begin{equation}
\intt{dy_0}|y_{01}|^{2\D-3}\vev{\f(x_1,y_0)\f(\xy{2})}=-\d(x_{12})\d(y_{12}).
\end{equation}

and the solution is,

\begin{equation}
\vev{\f(\xy{1})\f(\xy{2})}=c\, \d(x_{12})|y_{12}|^{1-2\D}.
\end{equation}

Notice that if using the path integral method to quantize the BMS free scalar model (<ref>), we will get the correlation functions with respect to the trivial vacuum instead of the highest weight vacuum <cit.>.

§.§ Localization of shadow integrals

In this subsection we discuss the localization of the shadow integrals. In the previous sections we notice that the integrals in the shadow formalism are localized to Dirac $\d$-distributions. The reason is similar to that of ambitwistor strings \cite{Mason:2013sva,Casali:2016atr,Casali:2017zkz}. We call the involved integrals as ``shadow integrals" with the following form
\begin{align}
\cA(x_{e},y_{e},\D_{n},\xi_{n})
& =\intt{\prod_{i\in I}{dx_i dy_i}}\prod\vev{\op\op}\vev{\op\op\op}\cdots\nn\\
& =\intt{\prod_{i\in I}{dx_i dy_i}}F_x(x_{n},\D_{n}) e^{F_y(x_{n},y_{n},\xi_{n})},
\label{eq:shadowIntegral}
\end{align}
where all the operators are virtual singlets $_n∈_n,ξ_n$, and $F_x$ is the collection of power functions and $F_y$ is the collection of exponential factors. Borrowing the terminology of Feynman diagram, we label the integrated positions by $i∈I$ and call them ``internal", $(x_i,y_i), i∈I$, and label the remaining positions by $e∈E$ and call them ``external", $(x_e,y_e), e∈E$. Since the exponential factors are bilinear with respect to $(y_n,ξ_n)$, we have
\begin{equation}
F_y=i A_0(x_n,y_e,r_n)+i\sum_{i\in I}y_i A_{i}(x_n,r_n),
\end{equation}
and the $y_i$-integration gives a $$̣-distribution localized on (the real points of) the algebraic variety

\begin{equation}
\cV[\cA]=\set{A_{i}(x_n,r_n)=\sum_{a\neq b}\frac{f_{ab}(r_n)}{x_{ab}}=0:i\in I}\subset \R^{2 |N|}\backslash\set{x_{ab}\neq 0:a,b\in N,a\neq b}
\label{eq:shadowVariety}
\end{equation}

where $f_{ab}$'s are linear functions of $r_n$. Then the integral (<ref>) simplifies to

\begin{equation}
\cA(x_{e},y_{e},\D_{n},\xi_{n})=(2\pi)^{|I|}\intt{\prod_{i\in I} dx_i \d(A_i)}F_x(x_{n},\D_{n}) e^{i A_0(x_n,y_e,r_n)}.
\label{eq:umbraIntegral}
\end{equation}

For convenience we refer to the variety $\cV:=\cV[\cA]$ associated with the shadow integral $\cA$ as “shadow variety". The shadow variety controls the behaviour of the shadow integral. We firstly summarize the properties of the shadow varieties appearing in the previous sections, and then briefly discuss the higher-point conformal blocks.

* Four-point conformal partial waves and blocks: $\cV$ can be solved as the equations of $x_{i},\,i\in I$, and each root corresponds to a block in the partial wave. The shadow variety $\cV$ may not have real points and then the integral vanishes.

dx_0 dy_0
_1_2 _0

* Bubble diagram: $\cV$ has two irreducible components, and the dimension of the intersection between $\cV$ with $\set{r_i=\text{const}:i\in I}$ is not zero, then there are nontrivial integrals of $x_i$ survived in (<ref>).

_1_2 _3
_1_2 _4,

{r_34/x_12-r_13,2/x_13+r_14,2/x_14=0,  r_34/x_12+r_23,1/x_23-r_24,1/x_24=0 }

* OPE block: the factor $F_x$ in the shadow integral depends on $y_{ab}$, then the remaining integral (<ref>) contains derivatives of $\d$-distribution on $$.
\begin{alignat}{2}
\intt{\dxy{0}}
\vev{\op_1\op_2 \opshadow_0} \op_{0},
\\
&& \frac{r_{01,2}}{x_{02}}+\frac{r_{02,1}}{x_{01}}=0.
\end{alignat}
\end{itemize}

\textbf{Higher-point conformal blocks.}
We briefly discuss the calculation of five-point conformal blocks via the shadow formalism. For the five-point there is only one type of OPE, named as the comb channel. The conformal partial wave with respect to exchanged singlets $_a,  _b$ can be constructed as
\begin{equation}
\CPWdressed_{a,b}(\xy{i})
\vev{\op_1\op_2\op_{a}}
\vev{\opshadow_{a}\op_{3}\op_{b}}
\vev{\opshadow_{b}\op_{4}\op_{5}},
\label{eq:fivepoint}
\end{equation}
and the shadow variety is
\begin{equation}
\cV=\{
\frac{r_{a1,2}}{x_{a1}}+\frac{r_{a2,1}}{x_{a2}}-\frac{r_{ab,3}}{x_{a3}}-\frac{r_{a3,b}}{x_{ab}}=0,\,
\frac{r_{b4,5}}{x_{b5}}+\frac{r_{b5,4}}{x_{b4}}-\frac{r_{ab,3}}{x_{b3}}-\frac{r_{a3,b}}{x_{ab}}=0
\}.
\end{equation}
The conformal partial wave contains four conformal blocks due to the shadow symmetry, implying that $$ as equations of $x_a,  x_b$ should have four roots, each of which contributes to a conformal block. After eliminating $x_b$ we indeed get a fourth order equation of $x_a$. However since the power factors are of the form $∏_i=0^5|F_i|^_i$, after inserting the solutions of $x_a,  x_b$, the result is unlikely to be simplified unless $_i∈$. We leave this possibility for further study.

For six-point conformal partial waves, there are two types of OPE, named as the comb channel and the snowflake channel. In both cases the shadow varieties $$ are defined by three equations. As equations of $x_i,  i∈I$, we check numerically that $$ has eight different roots, as expected from the shadow symmetry.

\section{Conclusion and Discussions}
In this work, we studied the shadow formalism for two-dimensional Galilean conformal field theory. As $2d$ Galilean conformal group is isomorphic to $3d$ Poincare group, we are allowed to use the Wigner-Mackey classification of $3d$ Poincare group to identify the unitary principal series representations and then to construct the shadow transform for GCFT$_2$. Using the shadow transform, we computed the OPE blocks, and discussed the Clebsch-Gordan kernels and the shadow coefficients. Moreover, we studied the conformal blocks and conformal partial waves in the framework of shadow formalism.

Furthermore we investigated several applications of shadow formalism, including the revisit of the decomposition of four-point function in Galilean MFT and the construction of a series of bilocal actions of Galilean MFT. In the revisit of the four-point function in MFT, we proposed a new inversion formula, due to different form of conformal partial waves from the one in \cite{Chen:2020vvn}. The resulting inversion function should be treated carefully and led to the correct conformal block expansion of four-point function. In constructing the bilocal actions of Galilean MFT, we used the intertwining bilinear forms, which obey the Ward identities. In special cases, these actions reduce to those of the BMS free scalar and the homogeneous tensionless fermionic string.

In our study, we came across a kind of integrals, which we called shadow integrals. The shadow integral can be reduced to the integral over an algebraic variety, due to the localization in the $y_i$ integration. This remarkable property makes analytic bootstrap in GCFT$_2$ feasible.

There are several future directions:

\textbf{More on shadow formalism.} In this work we mainly focused on the shadow formalism related to singlet representations, and the exchanged boost multiplets were dealt separately by the method of Casimir equations. Firstly it would be interesting to extend the current approach to boost multiplets in GCFT$_2$ and similarly to logarithmic multiplets in LogCFT. This requires a way to bypass the unitarity of the ``Euclidean" shadow transform. Secondly, when $ξ=0$ operators are involved in the shadow formalism there are technical difficulties to be settled. For example, the $ξ=0$ shadow transform is not unique due to the existence of different types of solutions of the Ward identities. Besides the shadow formalism, the $ξ=0$ subsector are found related to the celestial CFT recently \cite{Bagchi:2022emh}, and one may try to explore the role of the $ξ≠0$ subsector played in the celestial holography.

\textbf{GCFT$_2$ with/as defect.} In \cite{Setare:2011hc}, the lower-point correlation functions of boundary GCFT$_2$ were determined. Interestingly there are two types of boundaries with different residual symmetries, and in result the coincidence of $2d$ Galilean and Carrollian conformal symmetries can be distinguished. One could adopt the shadow formalism to the analytic studies of boundary and crosscap GCFT$_2$, and even interfaces between Galilean, Carrollian and relativistic CFTs. On the other side, the GCFT$_2$ itself can be regarded as the null defect in Lorentzian CFT$_3$, which deserves to be explored further.

\textbf{Deformations of MFT.} Constructing interacting GCFT$_2$ is an important task, and the numerical Galilean conformal bootstrap is obstructed by the lack of unitarity in GCFT$_2$.
The problem is that the highest weight representations contain negative-norm states except for the $ξ=0$ singlets, and if the spectrum contains only $ξ=0$ singlets, the theory reduces to a CFT$_1$. Hence a prior there is no positivity constraint on the bootstrap equations.
The unitary representations in the Wigner classification do not correspond to local operators, but are more interesting and can be related to the celestial/Carrollian correspondence, since the analogs in CFTs - the principal series representations have already appeared in the discussions of dS bootstrap and celestial CFTs.
In many aspects, the GCFT$_2$ in this paper is more similar to the logarithmic CFT. It would be interesting to add relevant interactions to the Galilean MFT and to investigate whether there are nontrivial fixed points, which will further serve as the target of the numerical Galilean conformal bootstrap.

\section*{Acknowledgments}
We would like to thank Peng-Xiang Hao and Zhe-Fei Yu for the participation in the early stage of this project. We are grateful to P. Hao, J. Qiao, Z. Yu, Y. Zheng for many valuable discussions.
We would like to thank L. Apolo, B. Czech, C. Chang, J. Lu, W. Song, J. Wu, G. Yang, W. Yang, X. Zhou for their stimulating questions in the workshops and seminars. The work is in part supported by NSFC Grant No. 11735001.

\vspace{1cm}
\appendix
\renewcommand{\appendixname}{Appendix~\Alph{section}}

\section{Conventions and Notations}\label{sec:app1}

\textbf{Notations in CFT$_1$ and GCFT$_2$:}
\begin{alignat}{2}
x_{12}& =x_1-x_2,
& \hs
k_{12}& =\frac{y_{12}}{x_{12}},
\\
\D_{12}& =\D_1-\D_2,
\xi_{12}& =\xi_1-\xi_2,
\\
x_{12,3}& =\frac{x_{13}x_{23}}{x_{12}},
k_{12,3}& =\frac{y_{13}}{x_{13}}+\frac{y_{23}}{x_{23}}-\frac{y_{12}}{x_{12}},
\\
\D_{12,3}& =\D_1+\D_2-\D_3, & \hs \xi_{12,3}& =\xi_1+\xi_2-\xi_3,\\
x_{12,34}& =x=\frac{x_{12}x_{34}}{x_{13}x_{24}},  & \hs
k_{12,34}& =\frac{y}{x}=\frac{y_{12}}{x_{12}}+\frac{y_{34}}{x_{34}}-\frac{y_{13}}{x_{13}}-\frac{y_{24}}{x_{24}},
\\
R_{i}& =\frac{\xi_{i}}{\xi_{0}},
R_{ij}& =\frac{\xi_{i}-\xi_{j}}{\xi_{0}}.
\end{alignat}

\textbf{Four-point configuration.} The permutation group $S_4$ acts transitively on the cross ratios of four points $(i), i=1,2,3,4$, and the stabilizer subgroup is the Klein four-group
\begin{equation}
V_4=\set{(1),(12)(34),(13)(24),(14)(23)}\simeq \Z_2\times \Z_2,
\end{equation}
that is, $g·x_12,34=x_12,34, g·k_12,34=k_12,34, ∀g∈V_4$. And effectively the action by $S_4/V_4 ≃S_3$ is shown in the table \ref{tab:fourpointcrossratio}.
\begin{table}[htbp]
\centering
\tablevspace{1.6}
\begin{tabular}{l|CCCCCC}
$S_3$ & (1) & (12)& (13) & (23) & (123) & (132)  \\
\hline
action & \frac{12\cdot 34}{13\cdot 24} & \frac{21\cdot 34}{14\cdot 23}& \frac{14\cdot 23}{13\cdot 24}& \frac{13\cdot 24}{12\cdot 34}& \frac{14\cdot 23}{21\cdot 34}& \frac{13\cdot 24}{14\cdot 23}\\
channel& s& & t& u& & \\
$x$ & x   & -\frac{x}{1-x}& 1-x& \frac{1}{x}& -\frac{1-x}{x}& \frac{1}{1-x} \\
$k$ & \frac{y}{x} & \frac{y}{x(1-x)} & -\frac{y}{1-x} & -\frac{y}{x} & -\frac{y}{x(1-x)} & \frac{y}{1-x}\\
$y$ & y& -\frac{y}{(1-x)^2}& -y & -\frac{y}{x^2} & \frac{y}{x^2}& \frac{y}{(1-x)^2}\\
\end{tabular}
\caption{The four-point cross ratios. The last three rows of cross ratios can be directly obtained by setting the four points to the conformal frame: $\set{0,x,1,\inf}$ in CFT$_1$ and $\set{(0,0),(x,y),(1,0),(\inf,0)}$ in GCFT$_2$.}
\label{tab:fourpointcrossratio}
\end{table}

\textbf{$3d$ Lorentzian/Euclidean conformal algebra.} In $^2,1$ or $^3$ we choose the coordinates $(x^0,x^1,x^2)$ and signatures $(±1,1,1)$. The generators of Lorentzian conformal algebra $(3,2)$ or its Euclidean partner $(4,1)$ are $M_ab,D,P_a,K_a$, $a=0,1,2$ and the corresponding vector fields are
\begin{equation}
p_{a}=\p_{a}, \hspace{3ex}
d=x^{a}\p_{a}, \hspace{3ex}
k_{a}=2x_{a}x^{b}\p_{b}-x^2\p_{a}, \hspace{3ex} m_{ab}=-x_{a}\p_{b}+x_{b}\p_{a}.
\end{equation}
The commutation relations are,
\begin{align}
& [M_{ab},M_{cd}]=g_{ad}M_{bc}+g_{bc}M_{ad}-g_{ac}M_{bd}-g_{bd}M_{ac},\nn\\
& [M_{ab},P_{c}]=-g_{ac}P_{b}+g_{bc}P_{a}, \hs [M_{ab},K_{c}]=-g_{ac}K_{b}+g_{bc}K_{a},\nn\\
& [D,P_{a}]=P_{a},\hs
\end{align}
and the default conjugation relation of the generators is anti-Hermitian, $Q^†=-Q$. The BPZ conjugation $D^†=D, P^†=K$ can be derived by switching between the N-S quantization and the radial quantization.

\subsection{Regularization of tempered distributions}\label{sec:distributions}

In the main text the calculations are undertaken in the framework of tempered distributions, and we have come across the regularization of power-type distributions. In this subsection we provide a mild introduction to the regularization and normalization of distributions by examples, following \cite{gel2014properties}. For simplicity the functions and distributions are on $$.

\textbf{Regularization.} A tempered distribution $∈'()$ acting on the rapidly decreasing test function $f∈()$ can be formally written as an integral $(f)=d x(x) f(x)$ with kernel $(x)$. One can imagine $f$ as a Gaussian wave-packet and $$ as some sharp observable. When the kernel $(x)$ is a function with singularities, e.g. $1/x-y$, the integral is convergent only for a small class $V$ of test functions $f∈V⊂()$. Then subtracting off all the divergent terms of the integral means extending the domain of $$ from $V$ to $()$. This procedure is called regularization of distributions.

For example, the regularization of $(x)=1/x-y$ can be chosen as
\begin{equation}
\PV\intt{dx}\frac{f(x)}{x-y}=\lim_{\e\to 0}\inttt{dx}{|x-y|>\e}\frac{f(x)}{x-y}.
\end{equation}
This is the Hilbert transform, also the shadow transform of $=$ in CFT$_1$ as reviewed in the appendix \ref{sec:cft1shadow}. It is a unitary operator on $L^2()$\footnote{The Hilbert space $L^2(\R)$ is canonically embedded into $\cS'(\R)$: $\cS(\R)\to L^2(\R)\to \cS'(\R)$, i.e. an example of rigged Hilbert space.} with two eigenspaces $H^2_+()⊕H^2_-()$, and as a result the (fermionic) principal series representation with $=$ of $SL(2,)$ is reducible.

The extension is usually not unique. We are interested in the case that a family of distributions $_a(x)$ depends on parameter $a$ analytically. Then the analyticity of $a$ helps us pick out a unique regularization of $_a(x)$. For example, the power-type distribution
\begin{equation}
(x_{+}^{a},f(x))=\inttt{dx}{(0,\inf)}x^a f(x),
\end{equation}
is convergent if $a> -1$ and hence is analytic with respect to $a$. If $a≤-1$ the integral is divergent and acquires regularization at $x=0$. Inserting a real-analytic test function $f(x)=∑_n=0^inf f^(n)(0)/n! x^n$ and interchanging the order of integration and summation, we have
\begin{align}
& =\inttt{dx}{(1,\inf)}x^a f(x) + \sum_{n=0}^{\inf} \frac{f^{(n)}(0)}{n!}\inttt{dx}{(0,1)}x^{a+n}\nn\\
& =\inttt{dx}{(1,\inf)}x^a f(x) + \sum_{n=0}^{\inf} \frac{f^{(n)}(0)}{n!}\frac{1}{a+n+1},
\end{align}
The first term is well-controlled due to the rapid decay of $f(x)$ and is irrelevant to our discussion. The second term implies that $(x_+^a,f(x))$ as a function of $a$ is meromorphic in $$, with simple poles at $a=-1,-2,…$ and residues $_a=-n (x_+^a,f(x)) = f^(n-1)(0)/(n-1)!$. Then stripping off the test function we find $x_+^a$ is meromorphic with respect to $a∈$, with simple poles at $a=-1,-2,…$ and residues,
\begin{equation}
\Res_{a=-n} x_{+}^{a} = \frac{(-1)^{n-1}}{(n-1)!}\d^{(n-1)}(0).
\end{equation}
Similarly $|x|^a,  |x|^a(x)$ and $x_-^a=þ(-x)|x|^a$ are all meromorphic with respect to $a∈$. We summarize their analytic structure in the table \ref{tab:distributions2}.

\begin{table}[htpb]
\centering
\renewcommand{\arraystretch}{1.6}
\begin{tabular}{L| L| L L}
\text{distributions}     & \text{poles}       & \text{residues} & \text{at}\\
\hline
x_{+}^{a}     & -1,-2,-3,\dots         & \frac{(-1)^{n-1}}{(n-1)!}\d^{(n-1)}(0)& a=-n\\

x_{-}^{a}   & -1,-2,-3,\dots          &\frac{1}{(n-1)!}\d^{(n-1)}(0) & a=-n\\

|x|^{a}       & -1,-3,-5,\dots       & \frac{2}{(2n)!}\d^{(2n)}(0) & a=-2n-1\\

|x|^{a}\sign(x) & -2,-4,-6\dots     & \frac{-2}{(2n-1)!}\d^{(2n-1)}(0) & a=-2n\\
\hline
\hline
\text{normalized ver.} & \text{removable poles}                     & \text{values}& \text{at}\\
\hline
\frac{1}{\G(a+1)}x_{+}^{a}   &-1,-2,-3,\dots  & \d^{(n-1)}(x) & a=-n\\

\frac{1}{\G(a+1)}x_{-}^{a}   &-1,-2,-3,\dots    & (-1)^{n-1}\d^{(n-1)}(x)& a=-n\\

\frac{1}{\G\left(\frac{a+1}{2}\right)}|x|^{a}& -1,-3,-5,\dots   & \frac{(-1)^n n!}{(2n)!}\d^{(2n)}(x) & a=-2n-1\\

\frac{1}{\G\left(\frac{a+2}{2}\right)}|x|^{a} \sign(x) & -2,-4,-6\dots  & \frac{(-1)^n (n-1)!}{(2n-1)!}\d^{(2n-1)}(x)& a=-2n\\
\hline
(x+i\e)^a& -1,-2,-3,\dots& x^{-n}-i\pi \frac{(-1)^{n-1}}{(n-1)!} \d^{(n-1)}(x)& a=-n\\

(x-i\e)^a& -1,-2,-3,\dots& x^{-n}+i\pi \frac{(-1)^{n-1}}{(n-1)!} \d^{(n-1)}(x)& a=-n\\
\end{tabular}
\caption{Homogeneous distributions on $\R$.}\label{tab:distributions2}
\end{table}

\textbf{Normalization of distributions.} We can cancel the simple poles of $x_+^a$ by a suitable Gamma function, and the normalized distribution $x_+^a/(a+1)$ is holomorphic with respect to $a$. Then the values at the removable poles are,
\begin{equation}
\frac{x_{+}^{a}}{\G(a+1)}\eval_{a=-n}=\d^{(n-1)}(x).
\end{equation}

Two other useful homogeneous distributions derived from boundary values of meromorphic functions are
\begin{equation}
(x+i\e)^a =x_{+}^{a}+e^{ia\pi}x_{-}^{a},
\hs
(x-i\e)^a =x_{+}^{a}+e^{-ia\pi}x_{-}^{a},
\end{equation}
and the poles are canceled at $a=-1,-2,…$ due to the factor $e^iaπ=(-1)^n$, hence they are holomorphic functions with respect to $a$. At the removed poles the values are
\begin{align}
(x+i\e)^{-n}& =x^{-n}-i\pi \frac{(-1)^{n-1}}{(n-1)!} \d^{(n-1)}(x),\nn\\
(x-i\e)^{-n}& =x^{-n}+i\pi \frac{(-1)^{n-1}}{(n-1)!} \d^{(n-1)}(x),
\end{align}
in which the distributions $x^-n$ are understood as $|x|^-2n$ and $|x|^-2n-1(x)$ respectively.

\textbf{Example.} As an example we provide the calculation of \eqref{eq:cft1doubleshadow} as follows,
\begin{align}
& =\lim_{\e\to 0}\inttt{dx_0}{\R} |x_{02}|^{-2(\D-\e)} |x_{01}|^{2\D-2}\nn\\
& = \lim_{\e\to 0}\pi\frac{\G(\D-\half)}{\G(1-\D)}\frac{\G(\half-\D)}{\G(\D)}\cdot \frac{|x_{12}|^{-1+2\e}}{\G(\e)}\nn\\
& =\frac{2\pi \tan\pi\D}{2\D-1}\d(x_{12}),
\label{eq:klt1}
\end{align}
where in the first line the integral is regularized by shifting the weight slightly, in the second line the $1d$ KLT integral \cite{Murugan:2017eto} is used, and in the last line the $$̣-distribution comes from regularization of the distribution $|x|^{\a}$. Notice that the KLT integral

\begin{equation}
\inttt{dx_0}{\R} |x_{01}|^{-2\D_1}|x_{02}|^{-2\D_2}=\pi^{\half}\frac{\G(\half-\D_1)}{\G(\D_1)}\frac{\G(\half-\D_2)}{\G(\D_2)}\frac{\G(\D_1+\D_2-\half)}{\G(1-\D_1-\D_2)}|x_{12}|^{1-2\D_1-2\D_2},
\end{equation}

is equivalent to the star-triangle relation by a special conformal transformation.

§ KINEMATICS AND SHADOW FORMALISM OF CFT_1

The $1d$ CFT appears in the SKY model, conformal defect and lightcone limit of higher dimensional CFT.
The discussion of GCFT$_2$ is similar to that of CFT$_1$ in many aspects.
In this appendix we review the kinematics of CFT$_1$, see e.g. <cit.>, including correlation functions, the shadow transform, OPE blocks and the conformal block expansion.

§.§ Local operators and correlation functions

The $1d$ Euclidean conformal group is $SO(2,1)$ with the generators $L_{n},\, n=\pm 1, 0$ obeying the commutation relations $[L_{n},L_{m}]=(n-m)L_{n+m}$, and to discuss fermionic representations, it should be replaced by the double cover $SL(2,\R)$. For simplicity we focus on the bosonic case. The conformal transformations on $\R^{1}$ are the fractional linear transformations as shown in the table <ref>. The primary operators at $x=0$ are defined by

\begin{equation}
[L_0,\op]=\D \op,\hs{3ex} [L_{-1},\op]=\p_{x}\op, \hs{3ex}[L_1,\op]=0,
\end{equation}

and the descendant operators are $\p^{n}_{x}\op$. The primary operators together with their descendants form a highest weight representation. When $\D> 0$ they are unitary irreducible representations of the Lorentzian conformal group $\wave{SL}(2,\R)$, named as the discrete series representations. The infinitesimal transformations of the primary operators $\op_{\D}(x)$ are

\begin{equation}
[L_n,\op(x)]=(x^{n+1}\p_x +(n+1)\D x^{n})\op(x),
\end{equation}

where $n=\pm1,0$, and the finite transformations are

\begin{equation}
U(f)\cdot \op(x)=|f'|^{\D}\op(x'),
\label{eq:globaltransformationscft1}
\end{equation}

where $x'=f(x)$ are the conformal transformations.

The $1d$ Lorentzian conformal group acting on the Lorentzian cylinder is the universal cover $\wave{SL}(2,\R)$. The covering map of these related groups are summarized as

\begin{equation*}
\wave{SL}(2,\R)
\overset{\Z}{\longrightarrow}
SL(2,\R)\simeq SU(1,1)\simeq Sp(1,\R)
\overset{\Z_2}{\longrightarrow}
PSL(2,\R)\simeq SO(2,1).
\end{equation*}

The $1d$ Euclidean conformal algebra is related to the Lorentzian one by the NS-quantization and Wick rotation,

\begin{equation*}
L_{0,E}=\frac{i}{2}(L_{-1,L}+L_{1,L}), \hs L_{\pm 1,E}=-\frac{i}{2}(L_{-1,L}-L_{1,L})\mp L_{0,L}.
\end{equation*}

name                    charge      vector field       finite transformation

translation         L_-1            _x                     x'=x+a

dilation                L_0             x_x           x'= x

SCT                 L_1             x^2_x      x'=x/(1-μx)

inversion           I        x'=-1/x

The generators of $1d$ conformal group. The last line is the inversion which is useful to check conformal covariance.

The two-point functions of primary operators $\op_{i}:=\op_{\D_i}$ are

\begin{equation}
\vev{\op_1 \op_2}=\d_{12}|x_{12}|^{-2\D}, \hs \D=\D_1=\D_2,
\end{equation}

and the three-point functions are

\begin{equation}
\vev{\op_1 \op_2 \op_3}=c_{123}|x_{12}|^{-\D_{12,3}}|x_{23}|^{-\D_{23,1}}|x_{31}|^{-\D_{31,2}},
\end{equation}

where $c_{123}$ is the three-point coefficient. The four-point function of $\op_{i}$ could be written as a product of the stripped four-point function $\fourpt^{\schannel}(x)$ and a kinematical factor $K^{\schannel}(x_i)$

\begin{equation}
\vev{\op_1 \op_2 \op_3 \op_4}=
\fourpt^{\schannel}(x).
\label{eq:schannel1}
\end{equation}

We choose the $\op_1\times \op_2\to \op_3\times \op_4$ kinematical factor as

\begin{equation}
\abs{\frac{x_{24}}{x_{14}}}^{\D_{12}}
\abs{\frac{x_{14}}{x_{13}}}^{\D_{34}},
\label{eq:kinematical1}
\end{equation}

then read the $\op_2\times \op_3\to \op_1\times \op_4$ kinematical factor by the permutation $(13)$

\begin{equation}
\vev{\op_1 \op_2 \op_3 \op_4}=
\abs{\frac{x_{24}}{x_{34}}}^{\D_{32}}
\abs{\frac{x_{34}}{x_{13}}}^{\D_{14}}
\fourpt^{\tchannel}(1-x).
\label{eq:tchannel1}
\end{equation}

The $s-t$ crossing equation from (<ref>) and (<ref>) is

\begin{equation}
x^{-(\D_1+\D_2)}\fourpt^{\schannel}(x)=(1-x)^{-(\D_2+\D_3)}\fourpt^{\tchannel}(1-x)\qas 0<x<1.
\end{equation}

§.§ Unitary principal series and shadow transform

The unitary irreducible representations of the Euclidean conformal group $SO(2,1)$ are classified into three classes: unitary principal series, discrete series and complementary series.

The Euclidean shadow transform is an intertwining map between two unitary principal series representations of the Euclidean conformal group. The unitary principal series representation $\urep{\D}$ of $SO(2,1)$ is defined as follows: the representation space is $L^2(\R)\ni f(x)$, with the inner product

\begin{equation}
\end{equation}

and the group action is the same as the one on the primary operators (<ref>) but with complex weight $(\D=\half+is),\, s\in\Rnot$,

\begin{equation}
U(f)\cdot \op(x)=|f'|^{\D}\op(x').
\end{equation}

Given a unitary principal series representation $\urep{\D=\half+is},\,s\in \Rneq$, we denote the associated shadow representation as $\urep{\wave{\D}=1-\D}$ and an operator transforming in $\urep{\wave{\D}}$ as $\opshadow$. The shadow transform $\shadow$

\begin{equation}
\shadow{\op}(x)
=\int_{\R}d x_0\,\vev{\opshadow(x)\opshadow(x_0)}\op(x_0)
=\int_{\R}d x_0\, |x-x_0|^{2\D-2}\op(x_0)
\end{equation}

is an intertwining map between two representations

\begin{equation}
\shadow: \urep{\D} \to \urep{1-\D}.
\end{equation}

If the representations $\urep{\D}$ and $\urep{1-\D}$ are both irreducible, by the Schur lemma, $\shadow$ is an isomorphism. To check this we apply the shadow transform twice $\shadow^2: \urep{\D} \to \urep{\D}$,

\begin{equation}
\int dx_1\, K(\shadow^2,x_1,x_2)\op(x_1)
=\int dx_0 dx_1\,\vev{\op(x_2)\op(x_0)}\vev{\opshadow(x_0)\opshadow(x_1)}\op(x_1).
\end{equation}

In the case that $\shadow$ is an isomorphism, the kernel of $\shadow^2$ should be proportional to identity, $K(\shadow^2)=\cN(\D) \delta(x_{12})$. This kernel $K(\shadow^2)$ can be evaluated via the KLT integrals and the prefactor is

\begin{equation}
\cN(\D)=\frac{2\pi \tan\pi\D}{2\D-1}.
\label{eq:cft1doubleshadow}
\end{equation}

When $\D=\half+is,\, s\in\Rnot$, the factor $\cN(\D)$ is finite and nonzero, hence the shadow transform is an isomorphism indeed.

At the position of the poles and the zeros of $\cN(\D)$, the operators $\shadow$ and $\shadow^{-1}$ are not isomorphisms. For example, when $\shadow$ is not injective, the kernel subspace $\ker \shadow$ is a sub-representation, and the maximal quotient $\urep{\D}/\ker \shadow{\urep{\D}}$ is the discrete series representation, see e.g. chapter 7 of <cit.>.

§.§ OPE blocks

The OPE relation of two primary operators is

\begin{equation}
\op_1(x_1)\op_2(x_2)=  \sum_{k} c_{12}^{k} \cD_{12k}(x_{12},\p_2) \op_{k}(x_2),
\label{eq:ope}
\end{equation}

and the bilocal operator $\cD_{12k}\op_k(x_2)$, capturing the resummation of the derivative operators, is called the OPE block <cit.>,

\begin{equation}
\cD_{12k}(x_{12},\p_{2})\op_k(x_2)
=x_{12}^{-\D_{12,k}}\Faa(\D_{k1,2}, 2\D_k ; x_{12} \p_{2})\op_k(x_2)
\label{eq:opeblock0}
\end{equation}

Here our convention follows the one used in <cit.>, and is slightly different from the one in <cit.>, which is

\begin{equation}
\op_1(x_1)\op_2(x_2)= x_{12}^{-\D_1-\D_2} \sum_{k} c_{12}^{k} \cB_{12k}(x_1,x_2).
\end{equation}

The relation between two conventions is

\begin{equation}
\cD_{12k}(x_{12},\p_2)\op_k(x_2) =x_{12}^{-\D_1-\D_2} \cB_{12k}(x_1,x_2).
\end{equation}

Notice that the technically safe way of phrasing the OPE relation (<ref>) is to introduce the vacuum OPE <cit.>,

\begin{equation}
\cR\, \op_1(x_1)\op_2(x_2)\vac= \sum_{k} c_{12}^{k} \cD_{12k}(x_{12},\p_2) \op_{k}(x_2)\vac
\end{equation}

where $\cR$ is the operator ordering with respect to a specific quantization scheme.

There are a few equivalent methods of computing the OPE block: the first is to use the compatibility of the OPE relation and the three-point functions, the second is to apply the recursion relation by imposing the symmetries on the OPE relation, and the third is to use the shadow formalism. We first recall the method of the recursion relation and then discuss the OPE blocks from the shadow formalism.

OPE blocks from recursion relations. Shifting the vacuum OPE to the origin and denoting the descendants as $\ket{\op_k,n}:=L_{-1}^{n}\ket{\op_k}$, we have

\begin{equation}
\op_1(x)\ket{\op_2}= \sum_{k} c_{12}^{k} \sum_{n=0}^{\inf} a_{n}(\op_k) x^{-\D_{12,k}+n} \ket{\op_{k},n}.
\end{equation}

The $L_{1}$'s action gives

\begin{align}
& =
[L_{1},\op_1(x)]\ket{\op_2}+\op_1(x) L_{1}\ket{\op_2}\nn\\
& =
(x^2\p +2\D_1 x)\op_1(x)\ket{\op_2}\nn\\
& =
\sum_{k} c_{12}^{k} \sum_{n=0}^{\inf} a_{n}(\op_k) (\D_{1k,2}+n)  x^{-\D_{12,k}+n+1} \ket{\op_{k},n},
\end{align}

and by using $L_1\ket{\op_k,n}=n(2\D_k+n-1)\ket{\op_k,n-1}$ we have

\begin{equation}
\sum_{k} c_{12}^{k} \sum_{n=0}^{\inf} a_{n}(\op_k) x^{-\D_{12,k}+n} n(2\D_k+n-1)\ket{\op_k,n-1}.
\end{equation}

Due to the orthogonality of the descendants we get the recursion relation of $a_n$

\begin{equation}
a_{n}(\op_k)=\frac{(\D_{1k,2}+n-1)}{n(2\D_k+n-1)}a_{n-1}(\op_k), \qand a_0(\op_k)=1,
\end{equation}

and the solution

\begin{equation}
a_n(\op_k)=\frac{(\D_{1k,2})_n}{(2\D_k)_n n!}
\end{equation}

matches with the Taylor coefficient of the hypergeometric function in (<ref>).

OPE blocks from shadow formalism. The operator product can be expanded at any other point $x_0\in [x_2,x_1],\,x_2<x_1$, and the translation invariance of the OPE block (<ref>) is not manifest. For example, to switch from $x_2$ to $x_1$ we may use the Kummer identity $\Faa(a,b;x)=\Faa(b-a,b;-x)e^{x}$,

\begin{align}
\cD_{12k}(x_{12},\p_2)\op_k(x_2)\vac
& =x_{12}^{-\D_{12,k}}\Faa(\D_{2k,1}, 2\D_k ; -x_{12} \p_{2})e^{x_{12} \p_{2}}\op_k(x_2)\vac\nn\\
& =\cD_{12k}(x_{21},\p_1)\op_k(x_1)\vac.
\end{align}

To make the translational invariance apparent, we can average over the expansion point $x_0$ by a weight function $f_{12k}$,

\begin{equation}
\cD_{12k}(x_{12},\p_2)\op_k(x_2)\vac=\inttt{dx_0}{I}f_{12k}(x_1,x_2,x_0)\op_k(x_0)\vac.
\label{eq:smearing1}
\end{equation}

The left hand side transforms as $\op_1(x_1)\op_2(x_2)\vac$, hence $f_{12k}\sim\vev{\op_1(x_1)\op_2(x_2)\opshadow_{k}(x_0)}$.
If insisting on the Euclidean region, there are extra unphysical contributions to the integral due to the shadow symmetry $\D\to1-\D$, and to single out the correct terms we need to introduce the projector of monodromy by hand <cit.>.

If Wick-rotating to the Lorentzian region, the three-point structure $\vev{\op_1(x_1)\op_2(x_2)\opshadow_{k}(x_0)}$ admits different analytic continuations, e.g. Wightman function and time-ordered function. For the time-ordered three-point structure, the OPE block (<ref>) contains shadow terms as the Euclidean case. The Wightman three-point structure is a more suitable choice and there are no shadow terms <cit.>. Equivalently, we can modify the integration region to eliminate the shadow terms. The weight function should vanish in the region where the OPE loses convergence, e.g., hitting other operators or becoming timelike with respect to the external operators. Hence in the two dimensional case, the integral domain is the causal diamond $\Diamond_{12}$ associated with spacelike pair of points $\set{x_1,x_2}$ <cit.>.

Comparing with the Taylor expansion (<ref>) of the OPE blocks, the integral expression (<ref>) corresponds to changing the basis of a conformal family. The original basis is  $\set{\ket{\op},L_{-1}\ket{\op},\dots }$, and the new basis is $\set{\op(x)\vac:x\in\R^{1}}$. They are related by the vacuum mode expansion,

\begin{equation}
\op(x)\vac=\sum_{n=0}^{\inf}\frac{x^{n}}{n!}\p^n \op(0)\vac
\end{equation}

We can check the equivalence of the two approaches by evaluating the integral (<ref>) directly,

\begin{equation}
\cD_{12k}(x_{12},\p_2)\op_k(x_2)=N_{12k} \inttt{d x_0}{I}\vev{\op_1(x_1) \op_2(x_2) \opshadow_k(x_0)} \op_k(x_0)
\end{equation}

where $N_{12k}$ is some normalization factor to be determined. In the CFT$_1$ case, the causal diamond degenerates to the interval $x_0\in (x_2,x_1)$ and we have

\begin{align}
\lhs
& =N_{12k} x_{12}^{-\D_{12,4}}\inttt{d x_0}{(x_2,x_1)}x_{02}^{-\D_{24,1}}x_{10}^{-\D_{41,2}} \op_k(x_0)\nn\\
& =N_{12k} x_{12}^{-\D_{12,4}}\inttt{d x_0}{(x_2,x_1)}x_{02}^{-\D_{24,1}}x_{10}^{-\D_{41,2}} \sum_{n=0}^{\inf}\frac{x_{02}^{n}}{n!}\p^n \op_k(x_2)\nn\\
& =N_{12k} \sum_{n=0}^{\inf}\p^n \op_k(x_2) I_n,
\label{eq:vacuumope}
\end{align}

where $\D_4=1-\D_k$ and the coefficient integrals are

\begin{equation}
=\frac{1}{n!}x_{12}^{-\D_{12,4}} \inttt{d x_0}{(x_2,x_1)}x_{02}^{-\D_{24,1}+n}x_{10}^{-\D_{41,2}}
= \frac{1}{n!} B(\D_{k1,2}+n,\D_{k2,1})x_{12}^{-\D_{12,k}+n},
\end{equation}

then by choosing $N_{12k}=B(\D_{k1,2},\D_{k2,1})^{-1}$ we get back to the previous result,

\begin{align}
\cD_{12k}(x_{12},\p_2)\op_k(x_2)
& =N_{12k}x_{12}^{-\D_{12,k}} \sum_{n=0}^{\inf}\p^n \op(x_2) x_{12}^{n}\frac{1}{n!} B(\D_{k1,2}+n,\D_{k2,1})\nn\\
& =x_{12}^{-\D_{12,k}} \Faa(\D_{k1,2}, 2\D_k ; x_{12} \p_{2}) \op(x_2).
\end{align}

§.§ Conformal blocks

Using the OPE relations repeatedly the higher-point functions can be decomposed into
a sum of the conformal blocks multiplied by the OPE coefficients. The conformal blocks
capture the contributions from the exchanged conformal families. This procedure has been explicitly done in the CFT$_1$ and CFT$_2$ case <cit.> for arbitrary higher-point. There are a few efficient ways of calculating four-point conformal blocks: solving the Casimir equations <cit.>, the recursion relations with respect to $\D$ <cit.>, and the shadow formalism <cit.>. We briefly illustrate the method of Casimir equation in CFT$_1$.

Settings of conformal block expansion. The conformal block expansion of four-point function is

\begin{equation}
\wick{\vev{\c1 \op_1 \c1 \op_2 \c1 \op_3 \c1 \op_4}} =\sum_{n}c_{12n}c_{43n}\cD_{12n}(x_{12},\p_2)\cD_{43n}(x_{43},\p_3)\vev{\op_n(x_2)\op_n(x_3)}=\sum_n p^{\schannel}_n \blockdressed^{\schannel}_n(x_i),
\end{equation}

where the (unstripped) conformal block with respect to primary $\op_n$ is defined as

\begin{equation}
\label{eq:block4}
\blockdressed^{\schannel}_n(x_i)=\cD_{12n}(x_{12},\p_2)\cD_{43n}(x_{43},\p_3)\vev{\op_n(x_2)\op_n(x_3)},
\end{equation}

and $p^{\schannel}_n=c_{12n}c_{43n}$. To further carry out calculations we need to introduce the stripped version of conformal blocks depending only on the cross ratios by factoring out the kinematical factors

\begin{equation}
\blockdressed^{\schannel}_n(x_i)=K^{\schannel}(x_i)\blockstripped^{\schannel}_{n}(x),
\label{eq:block1}
\end{equation}

then the block expansion of the stripped four-point function is

\begin{equation}
\fourpt^{\schannel}(x)=\sum_n p^{\schannel}_n \blockstripped^{\schannel}_{n}(x).
\end{equation}

Conformal blocks from Casimir equation. Inserting a complete basis into the four-point functions in the radial quantization $x_4>x_3>x_2>x_1$, we get

\begin{equation}
\vev{\op_1 \op_2 \op_3 \op_4}=\bra{0} \op_4 \op_3 \sum |\op_0|\, \op_2 \op_1\vac,
\end{equation}

where $|\op_0|$ is the projection operator of the conformal family $\rep{\D_0}$

\begin{equation}
|\op_0|=\sum_{n,m\in \rep{\D_0}}G^{-1}{}_{n,m}\ket{n}\bra{m},
\end{equation}

and $G_{n,m}=\bra{n}\ket{m},\, n,m\in \rep{\D_0}$ is the Gramian matrix. Then the conformal block can be written as a summation over the matrix elements $\bra{n}\op_1 \op_2 \vac$,

\begin{equation}
\label{eq:cft1block5}
\blockdressed^{\schannel}_0(\xy{i})=\bra{0} \op_4(x_4) \op_3(x_3) |\op_0| \op_2(x_2) \op_1(x_1)\vac.
\end{equation}

The Casimir differential operator is the representation of the abstract Casimir element of the conformal algebra when acting on the matrix elements $\bra{n}\op_1 \op_2 \vac,\, n\in \rep{\D_0}$. For element $X$ in the conformal algebra $\sllie(2,\R)$, the corresponding Ward identity is

\begin{equation}
\label{eq:Ward1}
\bra{n} X \op_1 \op_2 \vac
\end{equation}

where the differential operators are $X^{(i)}\vev{\dots \op_i(x_i)\dots }=\vev{\dots [X,\op_i(x_i)]\dots }$. The Casimir element of $\sllie(2,\R)$ is $C=L_0(L_0-1)-L_{-1}L_1$. Hence by repeatedly using (<ref>), and the fact that $C$ acts on $\rep{\D_0}$ as a scalar $\D_0 (\D_0-1)$, we find

\begin{equation}
\label{eq:Casimir5}
C^{(12)}\bra{n}\op_1 \op_2 \vac
=\D_0 (\D_0-1)\bra{n}\op_1 \op_2 \vac,
\end{equation}

where the differential operator $C^{(1+2)}$ is

\begin{equation}
C^{(1+2)}=-x_{12}^2 \p_1\p_2 -2x_{12}\ppair{\D_1 \p_2-\D_2 \p_1} +(\D_1 + \D_2)(\D_1 + \D_2-1).
\end{equation}

Inserting (<ref>) into the expression of conformal blocks, we get the Casimir equation

\begin{equation}
\ppair{C^{(1+2)}-\D_0 (\D_0-1)}\blockdressed^{\schannel}_{\D_0}(x_i)=0.
\end{equation}

Then plugging the definition of stripped conformal blocks into this equation, we get the Casimir equation

\begin{equation}
\bpair{x^2(1-x)\p^2_x +(\D_{12}-\D_{34}-1)x^2\p_x + \D_{12}\D_{34}x -\D_0 (\D_0-1)}\blockstripped^{\schannel}_{\D_0}(x)=0,
\end{equation}

which has two independent solutions: the physical block

\begin{equation}
\blockstripped^{\schannel}_{\D_0}(x)=x^{\D_0}\Fba(\D_0-\D_{12},\D_0+\D_{34};2\D_0,x),
\label{eq:cft1block}
\end{equation}

and the shadow block $\blockstripped^{\schannel}_{1-\D_0}(x)$.

§ GCFT_2 AS NULL DEFECT OF LORENTZIAN CFT_3

In this appendix we establish the relation between GCFT$_2$ and null defect in Lorentzian CFT$_3$ at the kinematical level, see also the discussion in higher dimensions <cit.> and a related discussion in <cit.>. We firstly recall the ideas of conformal defect, see e.g. <cit.> and the analytic studies in <cit.>, then discuss the residual symmetry of null defects in the first subsection, the outer automorphism of Carrollian conformal algebra in the second subsection. Notice that this defect picture rules out the infinite-dimensional BMS symmetry, and the result does not contradict to the symmetry enhancement argument in <cit.>, since the latter relies on the existence of local conserved currents, which a defect theory may not have.

Denoting the conformal group of the bulk[Not in the holographic sense.] CFT as $G$ and the one of the defect CFT as $G_{d}\subset G$, all the conformal defects connected by conformal transformations in $G$ can be regarded as points in the homogeneous space $G/G_{d}$, and the broken symmetries transforming one defect to another describe the motions in this homogeneous space. The defect can be regarded as a non-local object added to the original CFT spectrum, and besides the bulk-local operators describing local excitations in the bulk, there are defect-local operators describing local excitations on the defect. The dynamical information is captured by the bulk-bulk OPE, the bulk-defect OPE, the defect-defect OPE and the defect expansion, the last of which is a generalization of the Ishibashi states in $2d$ Virasoro CFT <cit.>.

Among the defects of various codimensions discussed in the literature, a trivial example is the point-like defect containing isolated points. For the one-point case, $G_{d}$ is the one-point stabilizer subgroup. If we insist that the state $\ket{\defect}$ describing the defect is $G_{d}$-invariant, in unitary CFT after modding out null states, $\ket{\defect}$ is translational-invariant and hence $\ket{\defect}=\vac$. Next considering the defect-local operators transforming in a non-trivial representation of $G_{d}$, together with their partners in the homogeneous space $G/G_{d}=\R^{d}_{c}$, where $\R^{d}_{c}$ is the conformal compactification of $\R^{d}$, we return to the construction of local operators.

For a Lorentzian CFT$_3$ with conformal group $SO(3,2)$, there are three types of $\codim-1$ conformal defects: timelike defect with residual symmetry $SO(2,2)$, spacelike defect with $SO(3,1)$ and null defect with $SO(1,1)\ltimes ISO(2,1)$, and the defect local operators should compose a Lorentzian CFT$_2$, a Euclidean CFT$_2$ and a special type of Carrollian CFT$_2$ respectively.

The residual conformal group of null defect $G_{d}=SO(1,1)\ltimes SO(2,1)$ is exactly the same as one-point stabilizer subgroup of local operators, and $G/G_{d}=\R^{2,1}_{c}$, indicating a relation between null defects and points in $\R^{2,1}_{c}$.
Apparently we can associate the null-cone $(x-X)\cdot (x-X)=0$ centered at $X\in \R^{2,1}$ with the point $X$ itself. To see this map from the null defect to the point preserving the symmetry, we consider the set of hyperboloids and hyperplanes in $\R^{2,1}$,

\begin{equation}
\M=\set{X^{+}x\cdot x+2X\cdot x+X^{-}=0 \,:\,X\in \R^{2,1} ,(X^{+},X^{-})\in\R^{1,1}}\simeq \R^{3,2}.
\end{equation}

In this set, the elements satisfying $X\cdot X-X^{+}X^{-}<0,>0,=0$ correspond to the timelike, the spacelike and the null defects respectively. Remarkably the conformal transformations of $SO(3,2)$ act linearly on $\M$.
Now the null-cone $X\cdot X-X^{+}X^{-}=0$ in $\M$ can naturally be identified with the embedding space of $\R^{2,1}$, i.e. $\R^{2,1}_{c}$.
In this way we establish a 1-1 symmetry-preserving correspondence between the null defects and the points in the conformal compactification of $\R^{2,1}$,

\begin{equation}
\text{null defect of }\R^{2,1}_{c} \hspace{3ex} \Longleftrightarrow \hspace{3ex} \text{point in }\R^{2,1}_{c},
\end{equation}

in which the null-planes correspond to the points at the conformal boundary of $\R^{2,1}$, and the conformal boundary itself corresponds to the spacelike infinity[Notice that this is not the asymptotic infinity. In the conformal compactification of $\R^{2,1}$, the timelike infinities and the spacelike infinity are identified.]. And as a byproduct we re-derive the embedding space formalism.

On the other side, the residual symmetries $ISO(2,1)$ act on the defect exactly as the Carrollian conformal transformations, hence we can describe the defect-local operators by Carrollian CFT$_2$. The extra $SO(1,1)\subset G_{d}$ can be identified with the non-trivial outer automorphism of the Carrollian conformal group $ISO(2,1)$. Despite that a Carrollian CFT may not possess this enhanced symmetry $SO(1,1)$, we can still learn the properties of the Carrollian CFT from the defect viewpoint.

§.§ Null-plane and null-cone defects

In this subsection we discuss three typical configurations of null defects: the null-plane, the lightcone and the conformal boundary. The settings of Lorentzian CFT$_3$ are summarized in appendix <ref>.

Null-plane. For the defect located at $x^{0}-x^{1}=0$, we use the lightcone coordinates, $y'=x^{0}-x^{1},\,y=x^{0}+x^{1},\,x=x^{2}$. Obviously the residual symmetry of the defect surface contains $x$-translation, $y$-translation and dilatation, and we can identify them with the generators of Carrollian conformal algebra, $L_{-1} =P_{2},\,M_{-1} =\half(P_{0}+P_{1}),\,L_{0} =D$. With the null vector $(1,1,0)$, the null rotation along the $x-y$ plane,

\begin{equation}
\exp \half v(M_{20}-M_{12})=\bmat{1+\frac{1}{8}v^2 &-\frac{1}{8}v^2 &\half v \\\frac{1}{8} v^2 &1-\frac{1}{8}v^2 &\half v \\\half v& -\half v& 1}
\end{equation}

preserves the defect as well. It acts as $x\to x,\,y\to y+vx$, hence suggesting that $M_{0}=-\half(M_{20}-M_{12})$.

To get the rest two generators, noticing that the action of the inversion $I: x^{0}\to \frac{x^{0}}{x\cdot x},\,x^{1}\to \frac{x^{1}}{x\cdot x},\,x^{2}\to \frac{-x^{2}}{x\cdot x}$ preserves the defect and acts as the inversion $x\to -\frac{1}{x},\,y\to \frac{y}{x^2}$ on the defect. Hence we can identify the SCTs as

\begin{equation}
L_{1}=K_{2}, \hs M_{1} =-\half(K_{0}+K_{1}).
\end{equation}

One can check that the six generators form a subalgebra $\isolie(2,1)\subset \solie(3,2)$. We summarize the identification of the generators in the following,

L_-1    = P_2,              L_0    = D,               L_1    = K_2,

M_-1    = (P_0+P_1),            M_0    = -(M_20-M_12),                M_1    = -(K_0+K_1).

The last residual symmetry is $D_{b}:=-M_{01}$ acting on the null-plane as $x\to x,\,y\to e^{-t}y$, and its commutation relations with other generators are

\begin{equation}
[D_{b},L_{n}]=0,     \hs  [D_{b},M_{n}]=-M_{n}.
\end{equation}

Lightcone. Another type of null defect locates at the lightcone $x\cdot x=0$. Apparently the Lorentz transformations $M_{ab}$ and the three SCTs, which reduce to

\begin{equation}
K: x^{a}\to \frac{x^{a}-t^{a}x^2}{1-2t\cdot x+t^2 x^2} \overset{x^2=0}{\longrightarrow}\frac{x^{a}}{1-2t\cdot x},
\end{equation}

preserve the lightcone, and should be identified with $L$'s and $M$'s in $\isolie(2,1)$ respectively.

The correct parameterization manifesting the Carrollian conformal symmetry in table <ref> turns out to be

\begin{equation}
(x^{0},x^{1},x^{2})=(\frac{1+x^2}{c y},\frac{1-x^2}{c y},\frac{2x}{c y}), \hs (\xy)\in \R^2,
\label{eq:lightcone1}
\end{equation}

where $c$ is an arbitrary non-vanishing constant.

The $x$-dependence in (<ref>) is inspired by the N-S quantization and the embedding space formalism of CFT$_1$. The Cayley tranform $z=\frac{1+ix}{1-ix}=\frac{1-x^2}{1+x^2}+i \frac{2x}{1+x^2}$ maps a CFT$_1$ on the real line $x\in \R^{1}$ to a CFT$_1$ on the unit circle $\th=2\arctan x \in S^{1}$, and from the embedding space viewpoint, this corresponds to choosing non-compact vs. compact slicing of the projective lightcone.

In our case the null direction of the lightcone is physical, and the parameterization such that $x$ transforms as linear fractional transformation is

\begin{equation}
\end{equation}

By comparing the actions of six residual symmetries on the defect with the Carrollian conformal transformations on the plane, we get a set of over-constrained equations of $F(\xy)$ with a unique solution $F(\xy)=\frac{(1+x^2)}{c y}$. Choosing $c=4$ we get the subalgebra inclusion,

L_-1    = -M_20+M_12,              L_0    = -M_01 ,              L_1    = M_20-M_12,

M_-1    = K_0-K_1,              M_0    = - K_2,                M_1    = K_0+K_1.

The bulk dilatation $D_{b}:=D$ acts on the null-cone as $x\to x,\,y\to e^{-t}y$, and its commutation relations are the same as before.

Conformal boundary. Notice that we can exchange the translations and SCTs in $\R^{2,1}$ by the inversion $I$, and the lightcone is mapped to the conformal boundary of $\R^{2,1}_{c}$. Then the subalgebra inclusion is

L_-1    = -M_20+M_12,              L_0    = -M_01 ,              L_1    = M_20-M_12,

M_-1    = P_0-P_1,              M_0    = -P_2,                M_1    = P_0+P_1,

and $D_{b}:=-D$. In the picture of embedding space of $\R^{2,1}$, $(X^{+},X^{-},X)=(1,x^2,x^{a})$, the lightcone defect is $X^{-}=0$ and the conformal boundary is $X^{+}=0$.

§.§ Bulk dilatation as outer automorphism of Carrollian CFT

In relativistic CFT$_d$, the semi-simplicity of the (complexified) conformal algebra $\mathfrak{so}(d+2,\C)$ implies that the outer automorphism group is finite. In Carrollian CFT$_2$, the outer automorphism group of both the global and local Carrollian conformal algebras is the multiplicative group, $\Out(\mathfrak{cca}_2)=\Rneq$. The connected component of identity in the group is generated by the bulk dilatation $D_{b}$, since there is no intrinsic scale in $\R^{2,1}$ mimicking the AdS radius.

From the null-cone defect picture, the bulk dilatation $D_{b}$ simply rescale the Carrollian time $y$. The infinitesimal action of $D_{b}$ is

[D_b,L_n]   =0,                [D_b,c_L]   =0,

[D_b,M_n]   =-M_n,                   [D_b,c_M]   =-c_M.

Defining the flowed generators as $Q(t)=U(t)QU(-t),\,U(t)=e^{t D_{b}}$, then we have

L_n(t)   = L_n,              c_L(t)   = c_L,

M_n(t)   =1/ M_n,                   c_M(t)   =1/ c_M.

where $\lm=e^{t}$. Defining the flowed highest weight state as $\ket{\D,\xi}_t=U_t\ket{\D,\xi}$ by $M_0(t)\ket{\D,\xi}_t=\xi \ket{\D,\xi}_t$, we find the boost charge

\begin{equation}
M_0 \ket{\D,\xi}_t  = \lm U_t M_0 \ket{\D,\xi}=\lm\xi \ket{\D,\xi}_t.
\end{equation}

Hence $D_{b}$ sends $\rep{\D,\xi,r}$ to a series of inequivalent ones $\rep{\D,\lm\xi,r}$. Either this series of operators is in the same theory, or there is a continuous series of theories labeled by $\lm$. If the theory is invariant under the outer automorphism, either there are only $\xi=0$ operators or the $\xi$-spectrum is continuous, and for a BMS field theory with this invariance, the central charge $c_{M}$ is forced to vanish, $c_{M}=0$.

Besides the multiplicative outer automorphism, there is a discrete automorphism. The discrete one is not in the identity component of the outer automorphism group, and it flips the sign of supertranslations

L_n'   = L_n,              c_L'   = c_L,

M_n'   =- M_n,                   c_M'   =- c_M.

§ MISCELLANEOUS CALCULATIONS

§.§ Calculation of the OPE blocks

In this appendix, we present some technical details in computing the OPE blocks. Inspired by the discussion in the appendix <ref>, we choose the integral region as $I= (x_2,x_1)\times \R, \, x_1>x_2$, then the integral (<ref>) can be evaluated as,

\begin{align}
\nn\\
& =
\inttt{d x_0y_0}{I}
\op_3(\xy{0})
\nn\\
& =
\inttt{d x_0y_0}{I}
\sum_{n,m=0}^{\inf}
\frac{x_{02}^{n}}{n!}\frac{y_{02}^{m}}{m!}
\p_{x_{2}}^n\p_{y_{2}}^m \op_3(\xy{2})
\nn\\
& =
\sum_{n,m=0}^{\inf}\p_{x_{2}}^n\p_{y_{2}}^m \op_3(\xy{2})
\label{eq:In1}
\end{align}

where $\D_4=2-\D_3,\,\xi_4=-\xi_3$. The coefficient integrals are

\begin{align}
& =
\frac{x_{12}^{-\D_{12,4}}e^{\xi_{12,4}k_{12}}}{x_{12}^{-\D_{12,3}}e^{\xi_{12,3}k_{12}}}
\inttt{d x_0y_0}{I}
\frac{x_{02}^{n}}{n!}\frac{y_{02}^{m}}{m!}
\label{eq:In}
\\
& =
\frac{1}{n!m!}
x_{12}^{-2\D_3+2}e^{2\xi_3 k_{12}}
\inttt{d x_0}{I}
\exp\bpair{\frac{\xi_{42,1}(-y_2)}{x_0-x_2}+\frac{\xi_{41,2}(y_1)}{x_1-x_0}}
\end{align}

where the $y_0$-integral is

\begin{equation}
\inttt{d y_0}{\R}
\exp\bpair{\ppair{\frac{\xi_{42,1}}{x_0-x_2}-\frac{\xi_{41,2}}{x_1-x_0}}y_0}.
\end{equation}

If (<ref>) is the correct ansatz for the OPE block, the integral (<ref>) should be a homogeneous polynomial of degree $(n+m)$ with respect to $x_{12}$ and $y_{12}$

\begin{equation}
\end{equation}

Let us look the first coefficient integral, which is related to the normalization factor and the shadow coefficient. The related $y_0$-integral is

\begin{equation}
\inttt{d y_0}{\R}
\exp\bpair{\ppair{\frac{\xi_{42,1}}{x_0-x_2}-\frac{\xi_{41,2}}{x_1-x_0}}y_0}.
\end{equation}

For virtual operators, the $\xi$'s are purely imaginary, hence this integral is proportional to a $\delta$-distribution

\begin{equation}
2\pi \d\bpair{\frac{2 \xi_3(x_0-X)}{(x_0 - x_2) (x_1 - x_0)}}
\frac{\pi}{|\xi_3|}(x_0 - x_2) (x_1 - x_0) \d(x_0-X),
\end{equation}

where the special point is

\begin{equation}
X=\frac{1}{2}(x_1+x_2)+\frac{R}{2} (x_1-x_2), \qand R:=\frac{\xi_1-\xi_2}{\xi_3}.
\end{equation}

The interesting thing is that the constraint $X\in (x_2,x_1)$ is equivalent to

\begin{equation}
\end{equation}

Thus, the first integral is

\begin{equation}
\label{eq:opeblocknormalization}
% =
% 2^{2-2\D_3}\frac{\pi}{|\xi_3|} \xi_3^{2-2\D_3}\xi_{312}^{-1+\D_{312}}\xi_{321}^{-1+\D_{321}}.
\end{equation}

For other coefficient integrals, the computation is straightforward. The higher $y_0$-integrals give rise to the derivatives of $\d(x)$

\begin{equation}
\frac{\pi}{|\xi_3|}(2\xi_3)^{-m} (x_{02}x_{10})^{m+1} \d^{(m)}(x_0-X) \exp(\frac{2 \xi_3(x_0-X)y_2}{(x_0 - x_2) (x_1 - x_0)}),
\end{equation}

where we have used the analytic continuation $\xi_n=ir_n\in i\R$. Inserting this into the expression of $I_{n,m}$ (<ref>), the integration with respect to $x_0$ turns into the derivatives of order $m$

\begin{align}
\label{eq:inm}
& =
\frac{(-1)^{m}}{n!m!}
\frac{\pi}{|\xi_3|}(2\xi_3)^{-m}
x_{12}^{-2\D_3+2}e^{2\xi_3 k_{12}}
\nn
\\
& \peq
\cdot \dv[m]{x_0}\eval_{x_0=X}
\exp(\frac{-\xi_{321} y_{12}}{x_{10}}).
\end{align}

For example, at the level $L=n+m=1$, there are two descendants $\set{\p_x\op,\p_y\op}$, with $2\times (L+1)$ coefficients

\begin{align}
\cD_{123}& \supset I_{1,0}(\xy)\p_x+I_{0,1}(\xy)\p_y\nn\\
& \supset \frac{1+R}{2} x \p_{x}+\ppair{-\frac{\D_{12}-\D_3 R}{2\xi_3}x+\frac{1+R}{2}y}\p_y.
\end{align}

We list the coefficients at level $2$ in table <ref>. In the table, the unlisted coefficient $a_{0,2,2}$ is

\begin{equation}
\frac{1}{16 \xi _3^2} \ppair{2 \D_{12}^2+\left(\D_3+1\right) \left(\left(2 \D_3+1\right) R^2-1\right)-2 \D_{12} \left(2 \D_3 R+R\right)}.
\end{equation}

L=2     x^2    xy    y^2

_x^2     1/8 (R+1)^2     0     0

_x_y      (R+1) (-2 _12+2 _3 R+R-1)/8 ξ_3     1/4 (R+1)^2     0

_y^2     a_0,2,2
(R+1) (-2 _12+2 _3 R+R-1)/8 ξ_3
1/8 (R+1)^2

The OPE block coefficients at level 2.

Along the diagonal direction the OPE block coefficients are the same up to a binomial coefficient, and this holds at the higher levels. This inspires us to make the simpler ansatz for the OPE block

\begin{equation}
\label{eq:opeblockgcft2}
\cD_{123}(x_{12},y_{12},\p_{x_2},\p_{y_2})=x_{12}^{-\D_{123}}e^{\xi_{123}\frac{y_{12}}{x_{12}}} \sum_{n,m} a_{n,m} (x_2\p_{x_2}+y_2\p_{y_2})^{n} (x_2\p_{y_2})^{m},
\end{equation}

in which the derivatives should be understood as acting on $\op_3$ directly, and the coefficients are related to the previous ones by

\begin{equation}
a_{n,m}=a_{n,m,n+m} \qiff I_{n,m}(x,0)=x^{n+m}a_{n,m}.
\end{equation}

Equivalently there is a recursion relation for $I_{n,m}$,

\begin{equation}
\label{eq:recursiongcft1}
x\p_y I_{n,m}(\xy)=(n+1)I_{n+1,m-1}(\xy) \qif m\geq 1,\,n\geq 0,
\end{equation}

which can be checked straightforwardly using (<ref>). Actually this is originated from the Ward identity with respect to $M_0$.

To calculate the leading coefficient $a_{n,m}$, from (<ref>) we read

\begin{equation}
\label{eq:opeblockgcft4}
\frac{(-1)^{m}}{n!m!}
\frac{\pi}{|\xi_3|}(2\xi_3)^{-m}
\dv[m]{x_0}\eval_{x_0=X}
\end{equation}

Here the derivative term can be expressed as a Jacobi polynomial. To be concrete, setting $z=R+2\frac{x_0-X}{x_{12}}$, then $x_{02}=\half (1+z) x_{12},\,x_{10}=\half (1-z) x_{12}$, and

\begin{align}
\peq \dv[m]{x_0}\eval_{x_0=X}
x_{02}^{\D_{31,2}+n+m-1}x_{10}^{\D_{32,1}+m-1} \nn\\
& =
\ppair{\frac{x_{12}}{2}}^{2\D_{3}+n+m-2}\dv[m]{}{z} \eval_{z=R}\bpair{(1-z)^{\D_{321}-1}(1+z)^{\D_{312}+n-1}(1-z^2)^{m}}\\
& =
\ppair{\frac{x_{12}}{2}}^{2\D_{3}+n+m-2}
\end{align}

Substituting it into (<ref>), we get

\begin{equation}
=x_{12}^{n + m}\cdot
\frac{2^{-n - m} \xi_3^{-m}}{n!}   (1 + R)^{n}   P^{(\D_{321}-1,\D_{312}+n-1)}_{m}(R).
\end{equation}

Hence the closed form of the OPE block is

\begin{align}
\cD_{123}(\xy,\p_{x},\p_{y})
\sum_{n,m} \frac{ (2\xi_3)^{-m}}{n!}
\ppair{\frac{1 + R}{2}}^{n}
\\
& \peq\cdot
(x\p_{x}+y\p_{y})^{n} (x\p_{y})^{m}.
\end{align}

In the above, we have used the definition and the Rodrigues' formula for the Jacobi polynomials

\begin{align}
P_{n}^{(a, b)}(z)
& =
\frac{(a+1)_{n}}{n !}\Fba(-n, 1+a+b+n ; a+1 ; \half(1-z))\\
& =
\frac{(-1)^{n}}{2^{n} n !}(1-z)^{-a}(1+z)^{-b} \dv[n]{z}\bpair{(1-z)^{a}(1+z)^{b}(1-z^{2})^{n}}
\end{align}

§.§ Relation between shadow coefficient and OPE block normalization

Consider the three-point structure $\vev{\op_1\op_2\shadow{\opshadow_3}}$, using (<ref>) and then inserting the OPE block (<ref>) into the three-point function $\vev{\op_1 \op_2 \op_3}$, we get

\begin{align}
\vev{\op_1\op_2\shadow{\opshadow_3}(\xy{4})}
& =\shadow(\op_1\op_2[\opshadow_3])\vev{\op_1\op_2\op_3(\xy{4})}\nn\\
& =\shadow(\op_1\op_2[\opshadow_3])N_{123}\int_I \vev{\op_1\op_2\opshadow_3}\vev{\op_3\op_3(\xy{4})}\label{eq:shadowco11}\\
& =\int_{\R^2} \vev{\op_1\op_2\opshadow_3}\vev{\op_3\op_3(\xy{4})},\label{eq:shadowco21}
\end{align}

where the last line comes from expanding the definition of $\shadow{\opshadow_3}$ in $\vev{\op_1\op_2\shadow{\opshadow_3}}$.

Notice that the integral regions in (<ref>) and (<ref>) are not the same. Assuming $x_1>x_2$, the OPE convergence condition implicitly used in (<ref>) is $x_4\notin (x_2,x_1),\,R\in(-1,1)$, and the OPE block integration is localized on $\set{X'}\times \R$, where

\begin{equation}
X'=\half(x_1+x_2)+\frac{R}{2}(x_1-x_2) \in (x_2,x_1).
\end{equation}

We can re-parametrize $x_4$ as

\begin{equation}
\end{equation}

then the former conditions are summarized as $[R,1,R_0,-1]$, where $[\dots ]$ denotes the cyclic order on the real projective line $\mathbb{RP}^{1}$. On the other hand, from (<ref>) the shadow coefficient integration (<ref>) is localized on

\begin{equation}
X = \frac{-(1-R)(1+R_0)x_1+(1-R_0)(1+R)x_2}{2(R-R_0)},
\end{equation}

and the relations are

\begin{align}
& X\in(x_2,x_1)\implies [R,1,R_0,-1] \qor [R_0,1,R,-1],\nn\\
& X\in(x_1,\inf)\implies [R,R_0,1,-1] \qor [R_0,R,-1,1],\nn\\
& X\in(-\inf,x_2)\implies [R_0,R,1,-1] \qor [R,R_0,-1,1].
\end{align}

Hence if the weights $\xi_i$ satisfy $R\in(-1,1)$, the two integral expressions (<ref>) and (<ref>) hold simultaneously and the normalization factor is

\begin{equation}
N_{123}^{-1}=\shadow(\op_1\op_2[\opshadow_3])= 2^{2\D_3-2}\frac{\pi}{|\xi_3|}(1-R)^{-1+\D_{23,1}}(1+R)^{-1+\D_{31,2}},
\end{equation}

matching with the result (<ref>).

§.§ Check of the second term in bubble integral

To check the result (<ref>), we calculate the second term in (<ref>) directly. Assuming $r_3r_4<0$, the $\d$-distribution localized on the variety \eqref{eq:secondvariety} is
\begin{equation}
\d(A_1)\d(A_2)=J_0\d(r_{3}+r_{4})\d\ppair{x_{2}-\frac{r_{13,2} x_1 x_3+r_{23,1} x_1x_4 -2 r_3 x_3x_4 }{r_3 \left(2 x_1-x_3-x_4\right)+r_{12} x_{34}}},
\end{equation}
where $J_0$ is a lengthy Jacobian factor
\begin{equation}
J_0=\frac{x_{12}^{2} x_{13} x_{23} x_{24}^{2}}{r_{13,2} x_{12} x_{23}^{2}+x_{24}^{2} \left(r_3 \left(3 x_1+x_2-4 x_3\right)-r_{12}x_{12}\right)}.
\end{equation}
Then we have
\begin{align}
& =A\,\shadow(\opshadow_1\opshadow_2[\op_4])^{-1}\inttt{dx_1}{\R}|x_{13}|^{-i(s_3+s_4)-1}|x_{14}|^{i(s_3+s_4)-1}\nn\\
& =A\,\shadow(\opshadow_1\opshadow_2[\op_4])^{-1}4\pi \d(s_3+s_4)\frac{1}{|x_{34}|}\nn
\\
& =\shadow(\opshadow_1\opshadow_2[\op_4])^{-1}\d(\op_3,\opshadow_4)\vev{\op_3(\xy{3})\op_3(\xy{4})},
\end{align}
where the prefactor is
\begin{equation}
A=2^{-1+\D_3+\D_4}\pi^3 |r_3|^{-4+\D_3+\D_4}|r_{23,1}|^{2-\D_3-\D_4}e^{2\xi_3\frac{y_{34}}{x_{34}}}|x_{34}|^{-1-\D_3+\D_4}.
\end{equation}
Notice that in the first line we cannot evaluate the integral by analytic continuation of $s_3,  s_4$  \eqref{eq:klt1} since the factor $A$ is singular at $x_34=0$. This justifies the result \eqref{eq:bubblelast} determined by shadow transform.

\subsection{Deriving the Casimir equations} \label{sec:derivingCasimirequations}

We may start from inserting a complete basis into the four-point functions in the radial quantization $x_4>x_3>x_2>x_1$,
\begin{equation}
\vev{\op_1 \op_2 \op_3 \op_4}=\bra{0} \op_4 \op_3 \sum |\op_0|\, \op_2 \op_1\vac,
\end{equation}
where $|_0|$ denotes the projection operator with respect to the conformal family $_0,ξ_0,r$
\begin{equation}
|\op_0|=\sum_{n,m\in \rep{\D_0,\xi_0,r}}G^{-1}{}_{n,m}\ket{n}\bra{m}.
\end{equation}
and $G_n,m=⟨n||m⟩,  n,m∈_0,ξ_0,r$ is the Gramian matrix of the inner product. In this way we can rewrite the conformal blocks \eqref{eq:block4gcft} as a summation over the matrix elements $⟨n|_1 _2 $, which are rescaled version of the three-point functions involving the descendants,
\begin{equation}
\label{eq:block5}
\blockdressed^{\schannel}_0(\xy{i})=\bra{0} \op_4(\xy{4}) \op_3(\xy{3}) |\op_0| \op_2(\xy{2}) \op_1(\xy{1})\vac.
\end{equation}
In principle we can use this projection operator to calculate the conformal blocks directly. When $ξ_0=0$ or $_0=0,-1,…$, the conformal family contains null states and we need to mod out them to get an invertible Gramian matrix \cite{Chen:2022jhx}. In this subsection we focus on the singlet case $r=1,ξ_0≠0$.

The Casimir differential operators are the representations of the abstract Casimir elements acting on the matrix elements $⟨n|_1 _2 $. For $X∈(2,1)$, the Ward identity is
\begin{align}
\bra{n} X \op_1 \op_2 \vac
& =
\bra{n}[X,\op_1(\xy{1})]\op_2(\xy{2})\vac+
\bra{n}\op_1(\xy{1})[X,\op_2(\xy{2})]\vac\nn\\
& =(X^{(1)}+X^{(2)})\bra{n}\op_1(\xy{1})\op_2(\xy{2})\vac,
\label{eq:Wardgcft1}
\end{align}
where the differential operators $X^(i)$ are from $X^(i)…_i(i)…=…[X,_i(i)]…$.

The Galilean conformal algebra $(2,1)$ admits two algebraically independent Casimir elements \eqref{eq:casimirPoincare}, and they act on the singlet $_0,ξ_0$ as scalars $_i$,
\begin{equation}
C_1=\lm_1=\xi_0^2, \hs
C_2=\lm_2=2\xi_0 (\D_0-1).
\label{eq:casimirsinglet}
\end{equation}
In other words, the Casimirs commute with the projection operators $C_i|_0|=|_0|C_i$. By repeatedly using \eqref{eq:Wardgcft1} we get
\begin{equation}
C^{(1+2)}_i\bra{n}\op_1 \op_2 \vac
\lm_i \bra{n}\op_1 \op_2 \vac,
\label{eq:Casimirgcft5}
\end{equation}
where the differential operators $C^(1+2)_i$ are of the forms
\begin{align}
& =
-x_{12}^2\frac{\p^2}{\p y_{1} \p y_{2}}+2x_{12}\ppair{\xi_1 \pdv{y_2}-\xi_2 \pdv{y_1}} +(\xi_1+\xi_2)^2,
\\
& =
2 x_{12} y_{12} \frac{\p^2}{\p y_{1} \p y_{2}}
+ x_{12}^{2}\ppair{\frac{\p^2}{\p x_{1} \p y_{2}}+\frac{\p^2}{\p x_{2} \p y_{1}}}
\nn\\
+ 2x_{12} \ppair{\D_1 \pdv{y_2}- \D_2 \pdv{y_1} - \xi_1 \pdv{x_2} +\xi_2 \pdv{x_1}}
\nn\\
&\peq  + 2y_{12} \ppair{ - \xi_1 \pdv{x_2} +\xi_2 \pdv{x_1}}
+ 2(\D_1+\D_2-1)(\xi_1+\xi_2).
\end{align}
Inserting \eqref{eq:Casimirgcft5} into the expression of the conformal blocks \eqref{eq:block5}, we get the Casimir equations,
\begin{equation}
\ppair{C^{(1+2)}_i-\lm_{i}}\blockdressed^{\schannel}_0=0,\qquad i=1,2.
\label{eq:singletCasimir}
\end{equation}
Then plugging the stripped conformal blocks \eqref{eq:block1gcft} into these equations, we get the Casimir equations of the stripped conformal blocks
\begin{equation}
\ppair{\cC_i-\lm_{i}}\blockstripped^{\schannel}_0(\xy)=0,\qquad i=1,2.
\end{equation}
To simplify the calculations we work in the slope coordinates $(x,k)=(x,y/x)$, and use the same symbol $g^(x,k)$ denoting the \schannel conformal block. In the slope coordinates we have
\begin{align}
\cC_1
& =
+(-\xi_{12}+\xi_{34}) x \pdv{k}
+\xi_{12} \xi_{34}x,
\\
\cC_2
& =
x k \pdv[2]{k}
+2x(x-1)\frac{\p^2}{\p x \p k}\nn
\\
& \peq
+\ppair{2+(-\D_{12}+\D_{34})x+(\xi_{12}-\xi_{34})x k }\pdv{k}
+(\xi_{12}-\xi_{34}) x^2 \pdv{x}\nn
\\
& \peq
+\ppair{\D_{12}\xi_{34}+\D_{34}\xi_{12}-\xi_{12}\xi_{34} k }x.
\end{align}

\subsection{Solving the Casimir equations}\label{sec:solvingCasimirEquations}
In this appendix we solve the Casimir equations and get the conformal block of singlet exchanged operator. For convenience we introduce the notation $R_ij≡ξ_ij/ξ_0$. Notice that under the shadow symmetry
\begin{equation}
(\D_0,\xi_0) \to (2-\D_0,-\xi_0), \hs{3ex} R_{ij}\to -R_{ij}.
\end{equation}
The Casimir equations are
\begin{align}
& \ppair{\cC_1-\lm_{1}}\blockstripped^{\schannel}_0(\xy)=0,\label{Casimir1}\\
& \ppair{\cC_2-\lm_{2}}\blockstripped^{\schannel}_0(\xy)=0.\label{Casimir2}
\end{align}
The first Casimir equation \eqref{Casimir1} can be solved by setting $F(x,k)=F_1(x)e^k F_2(x)$, and the solutions are
\begin{equation}
F(x,k)=c_1 f_{+}(x)\exp\bpair{\frac{k}{1-x}\ppair{\frac{\xi_{12}-\xi_{34}}{2}x+ h_{+}(x)}} + c_2 f_{-}(x)\exp\bpair{\frac{k}{1-x}\ppair{\frac{\xi_{12}-\xi_{34}}{2}x+ h_{-}(x)}},\label{solutionCasimir1}
\end{equation}
\begin{equation}
h_{\pm}^2(x)= \xi_{0}^2-(\xi_{0}^2+\xi_{12}\xi_{34})x +\frac{1}{4}(\xi_{12}+\xi_{34})^2 x^2.
\end{equation}
Before solving the second Casimir equation \eqref{Casimir2}, there are two subtleties to be stressed. Firstly, to manifest the shadow symmetry $ξ_0→-ξ_0, h_±→h_∓$, and to match the result with $ξ_12=ξ_34=0$, we need to choose the branch cut of $ξ_i$'s as
\begin{equation}
h_{\pm}(x)\equiv \pm \xi_0  H(x)= \pm \xi_0 \sqrt{1-(1+R_{12}R_{34})x +\frac{1}{4}(R_{12}+R_{34})^2 x^2}.
\label{eq:functionH}
\end{equation}
This does not change the solutions, and can be understood from the shadow formalism.
% Secondly, when the exchanged operator is degenerate, $\xi_0=0$, and $\xi_{12}\xi_{34}\neq 0$, $h(x)$ has unusual \schannel OPE behaviour $\sim \exp k(a x+b\sqrt{x})$. We deal with the case $\xi_0=0$ in [ref \tba].
Secondly, $H(x)$ contains two branch points with respect to $x$
\begin{equation}
x_{\pm}=2\frac{1+R_{12}R_{34} \pm \sqrt{(1-R_{12}^2)(1-R_{34}^2)}}{(R_{12}+R_{34})^2},
\label{eq:rootsOfHx}
\end{equation}
and the condition $x_±∉(0,1)$ rules out the following regions
\begin{equation}
R_{12},\,R_{34}>1, \qand R_{12},R_{34}<-1.
\end{equation}

Let us continue solving the Casimir equations. With the solution \eqref{solutionCasimir1}, the second Casimir equation \eqref{Casimir2} is reduced to a first-order differential equation of $f_±(x)$
\begin{equation}
\ppair{f'(x)/f(x)}_{\pm}=-\frac{ A_1(x)h_{\pm}(x) +A_2(x)}{8 (1-x)x h_{\pm}^2(x)},
\end{equation}
\begin{align}
& =
2(\D_{12}+\D_{34})(\xi_{12}+\xi_{34}) x^2
- 4\ppair{2(\D_{0}-1)\xi_{0}+\D_{34}\xi_{12}+\D_{12}\xi_{34}} x
+ 8(\D_{0}-1)\xi_{0},
\nn
\\
& =
(\D_{12}-\D_{34})(\xi_{12}+\xi_{34})^2 x^3
- 4(1+\D_{12}-\D_{34})(\xi_{0}^{2}+\xi_{12}\xi_{34})x^2
\nn
\\
& \peq + 4\ppair{(3+\D_{12}-\D_{34})\xi_{0}^{2}+\xi_{12}\xi_{34}} x
\end{align}
Hence the solutions of the two Casimir equations are
\begin{equation}
F(x,k)=c_1 \blockstripped_{+}(x,k) + c_2 \blockstripped_{-}(x,k),
\end{equation}
\begin{equation}
\blockstripped_{\pm}(x,k)=\exp\bpair{\frac{k}{1-x}\ppair{\half (\xi_{12}-\xi_{34})x \pm \xi_{0}H(x)}-\intt{dx}\frac{  A_1(x)h_{\pm}(x) +A_2(x)}{ 8(1-x)x h_{\pm}^2(x)}}.
\end{equation}

By checking the \schannel OPE limit $x, k→0$, and redefining the normalization to ensure the exchanged primary operator contributes one: $__0,ξ_0^∼x^_0 e^-k ξ_0 $, we find that the second solution $_-(x,k)$ can be identified to be the physical block
\begin{align}
\blockstripped_{\D_0,\xi_0}^{\schannel}(x,k)
& =
\frac{N(\D_0,\xi_0)}{H(x)}
\exp\bpair{\frac{k}{1-x}\ppair{\half (\xi_{12}-\xi_{34})x - \xi_{0}H(x)}}
\nn
\\
& \peq
\cdot x^{\D_0}
\bpair{\xi_{0}^{2}-\half(\xi_{0}^{2}+\xi_{12}\xi_{34}) x + \xi_{0}^{2}H(x)}^{1-\D_0}
\nn
\\
& \peq
\cdot \bpair{ \xi_{0}^{2}-\xi_{12}\xi_{34} + \half (2\xi_{0}^{2}- \xi_{12}^{2}- \xi_{34}^{2})x + (\xi_{12}-\xi_{34}) \xi_{0}H(x)}^{\half(\D_{12}-\D_{34})}
\nn
\\
& \peq
\cdot \bpair{ \xi_{0}^{2}+\xi_{12}\xi_{34} -\half (\xi_{12}+\xi_{34})^2 x + (\xi_{12}+\xi_{34}) \xi_{0}H(x)}^{\half(\D_{12}+\D_{34})}
\label{eq:strippedblock1}
\end{align}
where the normalization factor is
\begin{equation*}
N(\D_0,\xi_0)=2^{\D_0-1} \xi_{0}^{2\D_{0}-3} \bpair{(\xi_0+\xi_{12})(\xi_0-\xi_{34})}^{-\half(\D_{12}-\D_{34})} \bpair{(\xi_0+\xi_{12})(\xi_0+\xi_{34})}^{-\half(\D_{12}+\D_{34})},
\end{equation*}
and the other solution $_+(x,k)$ is proportional to the shadow block $_2-_0,-ξ_0^(x,k)$, thus the two solutions respect the shadow symmetry.

\subsection{Calculation of the projector of conformal partial waves}\label{sec:completeness1}
In this subsection we show that for four identical external operators the projection operator of conformal partial waves \eqref{eq:projector1} is proportional to
\begin{equation}
\cP^{1'2'3'4'}_{1234}\sim \th(1-x_{12,34})\d(x_{12,34}-x'_{12,34})\d(k_{12,34}-k'_{12,34}).
\label{eq:projector11}
\end{equation}
The overall coefficient is irrelevant to our discussion and has been omitted.
For the $_1_2_2_1$-type external operators, there seems no closed-form of this projector because the associated shadow variety reduces to higher order equations of $x_0,x'_0,y_0,y'_0$, and the remaining integration is hard to calculate.

Inserting the definition of conformal partial wave \eqref{eq:shadowCPW1} into the projection operator \eqref{eq:projector1} we have,
\begin{align}
&\peq \cP^{1'1'1'1'}_{1111}
\nn
\\
\inttt{\frac{dr_{0}ds_{0}}{4\pi^{2} \cN(\D_{0},\xi_{0})}}{\R\times \Rpositive}
\intt{dx_{0}dy_{0}dx'_{0}dy'_{0}}
\vev{\opshadow'_1\opshadow'_1\opshadow'_{0}}\vev{\op'_0\opshadow'_1\opshadow'_1}
\vev{\op_1\op_1\op_0}\vev{\opshadow_0\op_1\op_1}
\nn
\\
\inttt{\frac{dr_{0}ds_{0}}{4\pi^{2} \cN(\D_{0},\xi_{0})}}{\R\times \Rpositive}
\intt{dx_{0}dy_{0}dx'_{0}dy'_{0}}
\vpair{F_{0}}^{i\, s^0}
e^{i(A_0 y_{0}r_{0}+A'_0 y'_{0}r_{0}+A_{1}r_{0}+A_2)}
\nn
\\
&=\intt{dx_{0} dx'_{0}}
\d(A_0)\d(A'_{0})\d(A_{1})\d(F_{0})F_1 e^{iA_2},
\label{eq:projectorcalculation}
\end{align}
where $'_i=_i(x'_i,y'_i)$. In the second line the integral region of $s_0$ has been changed from $(0,inf)$ to $$ using \eqref{eq:inversionformula}, and we need to select the correct $$̣-distribution related to the conformal block in the partial wave. In the last line the substitution $r_{0}y_{0}\to y_{0},\,r_{0}y_{0}\to y_{0} $ is used to separate the first three terms in the exponential part, and the factor $r_{0}^2$ from $\cN$ gets canceled so that the integration of $y_{0},\, y'_{0},\, r_{0}$ gives three $\d$-distributions. The $s_0$-dependence is collected into $F_0$, and the integration gives $(̣F_0)$. The divergent volume factor is kept since we haven't done the gauge fixing procedure. One can check that after fixing $(x'_i,y'_i)$ to the standard conformal frame and renormalizing $'_1(x'_4,y'_4)$ as \eqref{eq:innerproductfourpoint}, the integral is finite.

In total the shadow variety $$ is defined by four equations,
\begin{alignat}{2}
&A_{0}:   \hs    && \frac{1}{x_{01}}+\frac{1}{x_{02}}-\frac{1}{x_{03}}-\frac{1}{x_{04}}=0,
\nn\\
&A'_{0}: &&             \frac{1}{x'_{01}}+\frac{1}{x'_{02}}-\frac{1}{x'_{03}}-\frac{1}{x'_{04}}=0,
\nn
\\
\frac{y_{1}}{x_{01}}+\frac{y_{2}}{x_{02}}-\frac{y_{3}}{x_{03}}-\frac{y_{4}}{x_{04}}
\nn
\\
\nn
\\
&F_{0}:&&\frac{x_{12}x_{30}x_{40}x'_{34}x'_{10}x'_{20}}{x_{34}x_{10}x_{20}x'_{12}x'_{30}x'_{40}}=\pm 1,
\nn
\end{alignat}
in which there are two irreducible components of $F_0$.

The rest integrals in \eqref{eq:projectorcalculation} can be done by solving $x_0,  x'_0$ from $A_0=A'_0=0$, giving rise to the overall factor in \eqref{eq:projector11}, then one irreducible component of $(̣F_0)(̣A_1)$ is proportional to $(̣x_12,34-x'_12,34)(̣k_12,34-k'_12,34)$.
The Heaviside function is originated from the reality condition of the shadow variety. The solution of $A_0=0$ is
\begin{equation}
\end{equation}
and the projection operator \eqref{eq:projectorcalculation} is nonvanishing only if the roots are real:
\begin{equation}
x_{13}x_{23}x_{14}x_{24}>0,\implies x_{12,34}<1.
\end{equation}

Another component of $(̣F_0)(̣A_1)$ is related to the first one by
\begin{equation}
x_{12,34}\to \frac{x_{12,34}}{x_{12,34}-1}, \hs y_{12,34}\to -\frac{y_{12,34}}{(1-x_{12,34})^2}.
\end{equation}
This is exactly the permutation symmetry of the four-point functions with identical external operators, and in result the projected four-point functions respect this symmetry automatically.

\newpage
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\field{title}{{Tensor representations of conformal algebra and conformally covariant operator product expansion}}
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