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Quote Polchinski String Theory volume 1 page 89.

$$S_f=\frac{1}{2\pi} \int d^2 z(b_{zz}\partial_{\bar z} c^z+b_{\bar z \bar z }\partial_zc^{\bar z})$$ Since the action... is weyl invariant, $b_{ab},c^a$ ... neutral under Weyl transformation. Since conformal transformation is a combination of a coordinate transformation $$z'=\sigma'^{1}+i\sigma '{}^2=f(z)$$ and a Wyel transformation. where the new metric is $$ds'^2=\exp(2\omega)|\partial_zf|^{-2}dz'd\bar z'$$ with $$\omega =\ln |\partial_z f|$$

For $h_b=2$ of $b_{zz}$ this might be understood from the $|\partial_zf|^{-2}$, but how to read off the $h_c=-1$ of $c^a$ from the action?

How to read of the conformal dimension of $bc$ CFT to be $(2,-1)$ from the action $S_g$?

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    $\begingroup$ Well you know by dimensional analysis that $h_b+h_c=1$. So if $h_b$ is 2, $h_c$ is -1. $\endgroup$ Sep 23, 2021 at 8:49

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