Sunday, 21 June 2020

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


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. 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.




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