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Partial answer : I think that the problem, in Polchinski book, is that the terms $g^{z \bar z} (=2)$, are systematically non explicitely written, that is, instead of : $\frac{1}{4 \pi}g^{z \bar z}$ …

answer for questions $1$,$2$, and $3a$ 1) Looking at $2.7.22$ to $2.7.24$ (and also $2.7.18abc$), one define the ghost number $N^g = \frac{-1}{2\pi i} \int_0^{2 \pi}dz :b(z)c(z):$ , and that all the …

I did not furnish all the details because it would be too long, but I give some hints at the end of the answer. I have used the formulae $:T^g: ~= ~:2(\partial c) b + c(\partial b):$ and $:\frac{1}{2 …

Bosonic path integrals : $$Z = \int D\phi ~e^{-i \large \int ~ dx [\frac{1}{2}\phi (\square+m^2)\phi]}$$ or Femionic path integrals (like Fadeev-Popov ghosts) : $$Z = \int D\eta D \tilde \eta ~e^{- …