I am learning conformal transformation, and this is by far the most confusing transformation for me.
For the 2D bc system
$$S=\frac{1}{2\pi}\int d^2 z b\overline{\partial}c,$$
we have the ghost current $j=-:bc:$. And $T(z)=:(\partial b)c: -\lambda \partial (:bc:)$. Their OPE is
$$T(z)j(0)\sim \frac{1-2\lambda}{z^3}+\frac{1}{z^2}j(0)+\frac{1}{z}\partial j(0)$$
On Polchinski's String Theory, in 2D CFT, under the infinitesimal conformal transformation $z^\prime=z+\epsilon v(z,\overline{z})$, we have for the corresponding ghost current the conformal transformation law
$$\epsilon^{-1}\delta j=-v\partial j-j\partial v+\frac{2\lambda -1}{2} \partial^2v.$$
Now, how do I get the Weyl transformation of $j$ based these information? Let's say under $\delta g_{ab}=2\delta \omega \delta_{ab}$ around the 2D flat Euclidean metric. (I am asking because I try to understand the relation between conformal transformation and Weyl transformation, which many resources claim they are related).
I would really appreciate it if someone can help me out of this confusion.
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3$\begingroup$ I don't think your question has anything to do with the bc system itself. You can relate $\epsilon v$ to $\delta\omega$ straightforwardly, see e.g. section 4.1 of Francesco et al's CFT textbook. $\endgroup$– octonionCommented Oct 11, 2021 at 6:17
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$\begingroup$ thanks! I will take a look at the resource you pointed. $\endgroup$– RuairiCommented Oct 11, 2021 at 7:41
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