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I've just started reading some introductory notes by Goddard and Gaberdiel on CFTs. The authors start by considering a Euclidean signature meromorphic field theory on the Riemann sphere. They state that

'...in the present context, the role of the Poincaré group is played by the group of Möbius transformations' (pg 11).

  1. Can someone please expand on this? How does the group of Möbius transformations enter the story?

  2. Do they mean that a QFT on $\mathbb{R}^2$ with Minkowski signature and Poincaré symmetry can be transformed into a QFT with Euclidean symmetry on the Riemann sphere with Möbius symmetry? If so, can this be explicitly shown?

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  1. The rest of the sentence in the quote on p. 11 explains it:

In conventional quantum field theory, the states of the theory transform in a suitable way under the Poincare group, and the amplitudes are therefore covariant under Poincare transformations. In the present context, the role of the Poincare group is played by the group of Moebius transformations ${\cal M}$, i.e. the group of (complex) automorphisms of the Riemann sphere.

  1. Morally: yes; literally: no. A globally-defined Wick-rotation is a delicate topic, which seems out of scope for these introductory notes. See also this related Phys.SE post.
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The authors are not suggesting that the Poincare group can be somehow transformed into the Mobius group. They are relating the study of CFTs to the study of QFTs which is likely to be more relatable to a reader of lecture notes. In QFT, the Hilbert space forms a representation of the Poincare group (which are the one-particle representations and tensor products thereof). Analogously, in a CFT the Hilbert space forms a representation of the Mobius group. The Mobius group enters here because it is the isometry group of the Riemann sphere $S^2$ (just like the Poincare group is the isometry group of ${\mathbb R}^{1,1}$).

There is no relation here between the Poincare and Mobius groups. The authors simply use the Poincare group as the QFT analogue of the Mobius group that shows up in a CFT.

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