I'm confused by the relation of coordinate transformation and conformal transformation. I found a nice note about conformal field theory written by David Tong. It does contain the demonstration related to my question, but I still don't understand. Here it goes,(adsbygoogle = window.adsbygoogle || []).push({});

The definition of conformal transformation is a coordinate transformation [tex]\sigma^a \rightarrow \sigma'^a(\sigma) [/tex], such that the metric changes by

[tex] g_{ab}(\sigma) \rightarrow \Omega^2(\sigma)g_{ab}(\sigma) [/tex]

What i'm confused is,Is the conformal transformation a subset of coordinate transformation?If this is true, then the conformal transformation is just part of the deffeomorphism(two-dimensional coordinate transformation is called deffeomorphism), but it seems that the answer is no.

Tong talked about the dynamical background metric and fixed background metric, he said if the metric is dynamical, the weyl transformation is deffeomorphism; however, if the metric is fixed, the weyl transformation is not deffeomorphism, but a physical symmetry that takes [tex]\sigma[/tex] to [tex] \sigma'[/tex].

And in a paragraph, he said, "any theory of 2d gravity which enjoys both deffeomorphism and Weyl invariance will reduce to a conformally invariant theory when the background metric is fixed." According to this, if the Weyl transformation is part of the deffeomorphism, then he could just say, ...enjoy deffeomorphism..., unnecessary to mentionbothdeffeomorhpism and Weyl invariance. (I take Weyl transformation same as conformal transformation.)

But, according to the definition, Weyl transformatio is evidently part of coordinate transformation. Anybody solves my confusion please?

Thanks for any instructions!

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# Coordinate transformation and conformal transformation?

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