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Is there any difference between signature $(1,1)$ and $(2,0)$ in 2D CFT?

The only thing I could thought of was that the previous one had Lorentz symmetry and the later one was Euclidean (rotation), but were they both 2D CFT? How do they differ? (such as two-point functions etc.)

Does study one of such structure the same as studying the other one? (i.e. by introduce an $iy$ rotation, but would there by issues?). Also, does that mean in 2D Euclidean space and Minkowski space was basically the same thing?

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  • $\begingroup$ Off the top of my head I would say Riemannian and pseudo-Riemannian manifolds don't have the same properties, e.g. the Hopf-Rinow theorem only holds for Riemannian manifolds. The signature here is important depending on what you're trying to do/compute. $\endgroup$
    – JamalS
    Jun 13, 2020 at 23:10

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Answer from TASI Lectures on Conformal Field Theory in Lorentzian Signature:

Unitary Lorentzian CFTs are related to reflection-positive Euclidean CFTs by Wick rotation. This is the Osterwalder-Schrader reconstruction theorem. (We describe the relationship in more detail below.) Thus, in principle, everything about a Lorentzian CFT is encoded in the usual CFT data (operator dimensions and OPE coefficients) that can be studied in Euclidean signature. However, many observables, and many constraints on CFT data are deeply hidden in the Euclidean correlators. Lorentzian dynamics provides a clearer lens to understand these observables and constraints.

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