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Maybe this question is rather too general to mention, but I hope the community can help to make it more precise in case it's needed.

The question is: if there's any link between the local conformal transformations of the Euclidean plane and the (local) diffeomorphisms of locally Lorentzian 4-manifolds?

In the answer to this question: Difference Between Algebra of Infinitesimal Conformal Transformations & Conformal Algebra, Qmechanic has alleged that the local (anti-)holomorphic maps on a Riemann sphere form a groupoid, isomorphic to the locally defined orientation-preserving conformal transformations.

Does it help with breaking down the (local) diffeomorphism group of the Lorentzian 4-manifold in terms of the local conformal group of the Euclidean plane?

Bastam Tajik
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    I don't know what "breaking down the local diffeomorphism group of a Lorenzian $4$-manifold in terms of the local conformal algebra in 2D" means, but there is a precise connection between a class of diffeomorphisms in asymptotically flat spacetimes (AFS) called superrotations and the 2D local conformal algebra. In fact, these superrotations are diffeomorphisms in AFS parameterized by conformal Killing vectors $Y^A(z,\bar z)$ on the celestial sphere at null infinity, and which act on such celestial sphere as its local conformal transformations. – Gold Aug 09 '22 at 23:49
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    See e.g. https://arxiv.org/pdf/1406.3312 and section (5.3) of https://arxiv.org/abs/1703.05448. – Gold Aug 09 '22 at 23:49
  • Very nice. The 2D space you referred to, is Euclidean or Minkowski? @Gold – Bastam Tajik Aug 10 '22 at 06:51
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    It is Euclidean. In AFS you have future and past null infinities ${\cal I}^\pm\simeq {\mathbb R}\times S^2$. The space called I referred to, called celestial sphere, is this $S^2$, which has Euclidean signature. If you want to understand the details I highly recomend you to study https://arxiv.org/abs/1703.05448. It is most certainly not what you had in mind, though. – Gold Aug 10 '22 at 13:15

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