conformal volume in this expression cancelling the one coming from the computation of residues in the worldsheet, or, equivalently, $$R^{j\omega}_{J}\rightarrow R_j.\nonumber$$ We thus find a complete agreement of (\[refle1\]) with (\[reflexion\])-(\[intertwining\]) in the unflowed limit.

Setting $j=j_1=j_2$ in (\[Kyalpha2\]) allowed us to compute both $\alpha$ and $B^{j\omega}_{J}$. However, we still need to check that these expressions are consistent with the terms proportional to $\delta(j_{12}^{+})$ in (\[Kyalpha2\]). Since $\alpha$ does not depend on $j$, the worldsheet and space-time integrations in (\[Kyalpha2\]) give the same result if, instead, we set $j=j_1=-1-j_2$. It follows that (\[Kyalpha2\]), and thus (\[Kyalpha\]), are satisfied as long as $$B^{j\omega}_{J}B^{-1-j,\omega}_{J}=-\frac{|1+2J|^2}{\pi^4} \frac{\gamma(2\Delta+2\omega(1+2J))}{\gamma(\omega(1+2J))^2\gamma(2\Delta)}.\nonumber$$ It is easily seen that the expression obtained in satisfies this identity.

Last but not least, we also need to check for the idempotence of the reflection symmetry defined by (\[refle1\]). Indeed, after applying (\[refle1\]) twice, this follows from the expression for the complex delta function, $$\label{deltacomp}
\delta(x_{12})=-\frac{|\epsilon|^2}{\pi^2} \int_{\mathbb C} d^2y\, \left|(x_1-y)^{-1+\epsilon}(y-x_2)^{-1-\epsilon}\right|^2,$$ used on both the worldsheet and the space-time integrations, together with $$R^{j\omega}_{J}R^{-1-j,\omega}_{-1-J}=1,\nonumber$$ an identity that straightforwardly follows from .

The contact term and the full propagator {#sect6}
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Eq. (\[refle1\]) can be used to compute the contact term of the propagator. For this, we take 