Why is it that every locally conformal transformation can be extended to a global conformal transformation for $D>2$?

Question

In $D=2$, we can have locally analytic transformations that cannot be globally well-defined.

However, for CFTs in $D>2$, we have only the global group. Why is that?

Also, is it a statement that depends on topology? Any references on that?

Answer 1

1. Let $\overline{\mathbb{R}^{p,q}}$ denote the conformal compactification of $\mathbb{R}^{p,q}$. Let $n:=p+q$ denote the dimension.

2. (The pseudo-Riemannian generalization of) Liouville's rigidity theorem states that if $n\geq 3$, then all local conformal transformations of $\overline{\mathbb{R}^{p,q}}$ can be extended to global conformal transformations. For a proof, see e.g. this Phys.SE post.

3. The upshot of Liouville's rigidity theorem is that for $n\geq 3$ there are only relatively few local conformal transformations of $\overline{\mathbb{R}^{p,q}}$ [which consist of (composition of) translations, similarities, orthogonal transformations and inversions], and these are easily extendable to global ones.

4. On the other hand, for $n=2$, there are many more local conformal transformations. See also e.g. this Phys.SE post.

5. Similar rigidity results (in the smooth case) hold on any conformal manifold $M$, cf. Wikipedia.