Polchinski's String theory Green's function on $RP_2$; eq. (6.2.38) p. 176 Is there an error in Polchinski's String Theory Equation (6.2.38) p 176 that is not on the Errata's page https://www.kitp.ucsb.edu/joep/links/joes-big-book-string/errata ?
He write for the Green's function on the Projective Plane $RP_2$
$$G'(\sigma,\sigma') = -\frac{\alpha '}{2} \log |z_1-z_2|^2 -\frac{\alpha '}{2} \log |1+z_1\bar{z}_2|^2. $$
The Projective Space is defined by identifying the point $z$ and $z' = -1/\bar{z}$, see (6.1.9). But the Green's function is not the same under that identification.
Performing the transformation $z\rightarrow -1/\bar{z}$ gives 
$$ G'(\sigma,\sigma')  \rightarrow -\frac{\alpha '}{2} \log \frac{|z_1-z_2|^2}{|z_1z_2|^2}-\frac{\alpha '}{2} \log \frac{|1+z_1\bar{z}_2|^2}{|z_1z_2|^2},
$$
which is not the same as the original Green's function. 
Is this a typo or am I missing something?
 A: I am pretty sure that this is not correct. Indeed GSW have for the Green's function on $RP2$ Eq. (8.3.21):
\begin{align}
G(z;z';q^2) = \eta^{\mu\nu} \left( \log |z-z'| + \log \left|\frac{q^2}{z\bar{z}'} +1\right| \right)
\end{align}
where the points with $z$ and $-q^2/\bar{z}$ are identified. In Polchinski's convention this would correspond to 
\begin{align}
G_\mathrm{GSW} (\sigma_2,\sigma_2) = -\frac{\alpha'}{2} \log |z_1-z_2|^2  -\frac{\alpha'}{2} \log\left|\frac{1}{z_1\bar{z}_2} +1\right| ^2 
\end{align}
Under $z_1\rightarrow -1/\bar{z}_1$ this now transforms as
\begin{align}
G_\mathrm{GSW} (\sigma_2,\sigma_2) &=  -\frac{\alpha'}{2} \log (z_1-z_2)(\bar{z}_1-\bar{z}_2) \left( \frac{1}{z_1\bar{z}_2} +1\right)\left( \frac{1}{\bar{z_1}z_2} +1\right)\nonumber\\
& \rightarrow -\frac{\alpha'}{2} \log \left(-\frac{1}{\bar{z}_1} - z_2 \right) \left(-\frac{1}{z_1} -\bar{z_2}\right)  \left( -\frac{\bar{z}_1}{\bar{z}_2} +1\right)\left(-                  \frac{z_1}{z_2} +1\right) \nonumber\\
&= -\frac{\alpha'}{2} \log \frac{(1+\bar{z}_1z_2) (1+z_1\bar{z}_2) (\bar{z}_1-\bar{z}_2)(z_1-z_2)}{\bar{z}_1 z_1 \bar{z}_2 z_2}\nonumber\\
&=  -\frac{\alpha'}{2} \log \left( \frac{1}{\bar{z_1}z_2} +1\right) \left( \frac{1}{z_1\bar{z}_2} +1\right) (\bar{z}_1-\bar{z}_2)(z_1-z_2) \nonumber\\
&= G_\mathrm{GSW} (\sigma_2,\sigma_2) 
\end{align}
Both Greens functions are related:
\begin{align}
G'(\sigma_1,\sigma_2) = G_\mathrm{GSW} (\sigma_2,\sigma_2)  -\frac{\alpha'}{2} \log |z_1 z_2|^2
\end{align}
