I kinda forgot about this thread for a while, but here is one error that I just noticed. In page 337, while working out the partition function for a (Euclidean) CFT on a torus, they write the translation operator that translates by a full period in the direction of ## \omega_2 ## as:
$$ \exp [-(H \cdot Im \omega_2 -iP\cdot Re\omega_2)]. $$
This is correct, BUT then they write the momentum operator on a cylinder with circumference ##L## as
$$P = \frac{2\pi i}{L} (L_0-\bar{L}_0) .$$
This is WRONG, the correct expression is without the imaginary ##i##. See for instance the notes of Cardy on CFT and statistical physics, or Polchinski page 209, or Ginsparg page 83 and subsection 2.2. The reason is that the dilation generator on the plane is ##D=L_0+\bar{L}_0##, while the rotation generator is ##\mathcal{P}=i(L_0-\bar{L}_0)##. Now consider the case of a cylinder, let's say one with ##L=2\pi## so that we can drop all the constants. Finite translation along the axis of the cylinder (Euclidean time evolutions) are given by ##e^{-D}##, while finite translations around the cylinder are given by ##e^{-\mathcal{P}}##. However, the convention we used above dictates that ##\mathcal{P}## is not the momentum. Rather, if we want to use the form for the translation operator used by the Yellow Book above, the momentum is ##P=-i\mathcal{P}=L_0-\bar{L}_0##, and everything works out fine. Bottom line is, the correct expression should be:
$$P = \frac{2\pi }{L} (L_0-\bar{L}_0) .$$
I included this whole explanation in case someone read this in the book and went to Google to figure out what's wrong.