# 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?

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}