3
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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}

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.