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I think the original source of this claim is the famous unpublished paper of Luescher and Mack. Everyone's citing it. It is more rigorous mathematically and general (they don't assume parity) than Di Francesco. It starts on pages 1-2 of the manuscript. The proof below is basically the same proof, just with added details and a little bit different notation. ...

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This extra factor arises from the analogy of the conformal factor $\alpha'\omega$ term in (6.2.16). The required $\omega$ is $$\omega = \ln \left ( \frac{2\pi}{\partial_\nu\vartheta_1}\right)$$ and substituting it to the exponential we get $$\exp\left( -\frac{\alpha'}{2}\sum_ik_i^2 \cdot \ln \frac{2\pi}{\partial_\nu\vartheta_1} \right) = ... 2 Comments to the question (v2): To be specific, let us assume that the underlying 2D manifold is the Riemann sphere S^2\cong \mathbb{C}\cup\{\infty\}. The group of globally defined conformal transformations is the 6-dimensional group PSL(2,\mathbb{C}) of Moebius transformations. Mathematically speaking, one should consider the groupoid of locally ... 1 For clarity let's work with a Lorentzian signature. Our g is a metric for a 2 dimensional Lorentzian manifold M. It is well-known that any two dimensional pseudo-Riemannian manifold is conformally flat, that is$$g = e^{2\omega}\eta Where $\eta$ is the flat 2D Minkovski metric. Your Lightcone gauge example You didn't define your $x^\pm$s but I ...

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I recently stumbled upon a good comment about this in Jared Kaplan's AdS/CFT notes Any quantum field theory which has hope of having an UV-completion can be viewed as as effective theory at point in the RG flow from an UV complete theory. Field theories at the UV fixed point are conformal. Hence all 'well-defined' field theories are either CFTs or points ...

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This is a very broad question and therefore impossible to answer completely, but I will try to answer some questions and refer to some literature. Your statement of the AdS/CFT correspondence was not quite complete: Type IIb string theory on asymptotically $AdS_5\times S^5$ is equivalent to $\mathcal{N}=4$ super Yang-Mills theory In the limit of large ...

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