# How is the $d=2$ Weyl invariance different from the generic Conformal $SO(2,d)$ invariance?

First of all, I read this answer and I understand that Weyl transformations are transformations of the metric and Conformal transformations are transformations of the coordinates that "Weyl-transform" the metric, but from what I see Weyl and Conformal invariance are pretty much the same thing: doing something that only changes the metric by a positive rescaling.

In my String Theory course, we talked about the $$d=2$$ worldsheet of the string being Weyl-invariant, and how this invariance (particularly in $$d=2$$) is important. Later on, we moved to AdS/CFT, and studying the properties of a CFT we mentioned that it is conformally invariant (duh) but since it generally is in $$d>2$$ it won't be "as nice as" the $$d=2$$ invariance of String Theory. Why is that? What changed?

• Ok, but what does this imply? What can I do in $d=2$ that I can't do in $d>2$? – Mauro Giliberti Jun 3 at 14:09
• To be completely correct one should say that the symmetry algebra becomes infinite dimensional. The global conformal group is a bit special in $D=2$ – NDewolf Jun 3 at 15:14