affine vertices. We shall argue later about some conformal symmetries they retain.

The transformation between the space-time picture and the $m$-basis is carried out in analogy with (\[mellin\]), namely, $$\label{fourierJM}
 \Phi^{j\omega}_{J}(M|z) = \int_{\mathbb C} d^2x \left|x^{J+M}\right|^2 \Phi^{-1-j,\omega}_{-1-J}(x|z),$$ where $M$ is the $J^3_0$ eigenvalue. By means of this map, we have $$\label{identif33}
 \Phi^{j\omega}_{J}(\pm J|z) \propto \Phi^{\mp\omega}_{j}(\pm J \pm k\omega/2|z).$$ If the normalization of $\Phi^{j\omega}_{J}(x|z)$ is defined so that the the relation in (\[identif33\]) is actually an identity, this can then be used for determining the regular term of the propagator in the $x$-basis. The dependence of the amplitude on the boundary coordinates is fixed once invariance under the global $\mathfrak{sl}(2,\mathbb C)$ symmetry is imposed, so that $$\left\langle \Phi^{j_1\omega_1}_{J_1}(x_1|z_1) \Phi^{j_2\omega_2}_{J_2}(x_2|z_2)\right\rangle\propto\delta^2(J_{12})\left|x_{12}^{2J_1}\right|^2,\nonumber$$ for $J_1\sim J_2$. Transforming this expression by means of (\[fourierJM\]) we obtain $$\left\langle \Phi^{j_1\omega_1}_{J_1}(M_1|z_1) \Phi^{j_2\omega_2}_{J_2}(M_2|z_2)\right\rangle \propto \pi \delta^2(J_{12}) \delta^2(M_1+M_2) \frac{\gamma(1+J_1-M_1)\gamma(-1-2J_1)}{\gamma(-J_1-M_1)},\nonumber$$ where the proportionality constant depends on all the parameters, with the exception of the spin projections. This overall factor can thus be computed by setting $M_1=-J_1$ and $M_2=J_2$, after using (\[identif33\]), together with (\[2ptsl2bb\]). We obtain $$\begin{aligned}
\label{regular}
&& \left\langle \Phi^{j_1\omega_1}_{J_1}(x_1|z_1) \Phi^{j_2\omega_2}_{J_2}(x_2|z_2)\right\rangle=\nonumber \\
&& ~~~~~ ~~~~~ -\frac{|1+2J_1|^2}{\pi^2}\delta_{\omega_1\omega_2}\delta^2(J_{12})\left[\delta\left(j_{12}^+\right)+Y^{j_1\omega_1}_{J_1}\delta\left(j_{12}\right)\right]\left|x_{12}^{2J_1}z_{12}^{-2\Delta_1}\right|^2,\end{aligned}$$ where $$Y^{j\omega}_{J}=\frac{\pi B_j}{\gamma(-2j)}\frac{\gamma_{\omega}(-1-j-J)}{\gamma_{\omega}(j-J)}, \qquad \qquad
 \gamma_{\omega}(x)=\gamma(x+k\omega/2),\nonumber$$ and $$\Delta=\Delta_0-\omega-\omega J+\frac{k}{4}\omega^2.\nonumber$$ In the unflowed limit, namely, for 