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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?

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Conformal invariance in two dimensions is special because the symmetry group is much much larger in D = 2. In higher dimensions the conformal group is SO(D+1, 2) which is finite dimensional. But in D = 2 you find that conformal transformations are just holomorphic and anti-holomorphic coordinate transformations which gives you an infinite set of local coordinate transformations. This gives D = 2 conformal theories much more restrictions.

The subject of 2 dimensional conformal field theories is extremely rich and complex, I recommend Di Francesco's book on CFTs as the subject is much too large to explain here.

Basically the symmetry properties of 2D CFTs mean that your observables (Correlation functions for example) must obey a much larger set of relations (because of the larger symmetry group). This means that given some initial data (like the scaling dimensions of your fields/operators) one can fix an extremely large set of the observables without having to do basically any calculations.

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    $\begingroup$ Ok, but what does this imply? What can I do in $d=2$ that I can't do in $d>2$? $\endgroup$ – Mauro Giliberti Jun 3 at 14:09
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    $\begingroup$ To be completely correct one should say that the symmetry algebra becomes infinite dimensional. The global conformal group is a bit special in $D=2$ $\endgroup$ – NDewolf Jun 3 at 15:14
  • $\begingroup$ @MauroGiliberti I've edited my post $\endgroup$ – CStarAlgebra Jun 3 at 16:39

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