# Equivalent definitions of primary fields in CFT

I have come across two similar definitions of primary fields in conformal field theory. Depending on what I am doing each definition has its own usefulness. I expect both definitions to be compatible but I can't seem to be able to show it. By compatible I mean definition 1 $\iff$ definition 2. I will write both definitions in the two-dimensional case and restricting to holomorphic transformations.

Def #1 from Francesco CFT: A field $f(z)$ is primary if it transforms as $f(z) \rightarrow g(\omega)=\left( \frac{d\omega}{dz}\right)^{-h}f(z)$ under an infinitesimal conformal transformation $z \rightarrow \omega(z)$.

Def #2 from Blumenhagen Intro to CFT: A field $f(z)$ is primary if it transforms as $f(z) \rightarrow g(z)=\left( \frac{d\omega}{dz}\right)^{h}f(\omega)$ under an infinitesimal conformal transformation $z \rightarrow \omega(z)$.

Can someone show me how they are indeed the same?

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Cross-posted to mathoverflow.net/q/105489 – Qmechanic Mar 25 '13 at 22:16

$${dz\over dw} = ({dw\over dz})^{-1}$$
I don't think that is all b/c then the transformation is $f(\omega) \rightarrow g(\omega) = ...$ which is different from the first definition. – yca Aug 23 '12 at 17:42
The first definition is a relation between the new field $g$ evaluated at the new point and the old field $f$ at the the old point. The second definition is a relation between new field at the old point and the old field at the new point. Simply changing the names of the variables in one of the definitions does not give me the other. – yca Aug 24 '12 at 13:43