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

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 $$S_g$$?

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