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The conformal transformation $g'_{\mu\nu} = e^{-2\sigma}g_{\mu\nu}$, $sigma = sigma(x)$ leads to the transformation of the Ricci scalar $$R' = e^{2\sigma}R - 12e^{2\sigma}(2\sigma_{,\mu}^{,\nu} - 2\sigma_{,\mu}\sigma^{,\mu}$$ Since $\phi' = e^{\sigma}$ then  \frac{1}{12}\phi'^2R' = \frac{1}{12}\phi^2 R - \phi^2(2\square\sigma - ...

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This is standard material for any text on 2d CFT's. First, the Virasoro algebra contains holomorphic and anti-holomorphic parts and you can see (exercise!) that the conformal blocks factorize into respective pieces. So lets consider just the holomorphic part. The key is the decomposition of the identity in the conformal module of the intermediate state (the ...

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TQFTs by definition satisfy cutting and gluing axioms. Roughly speaking, you should be able to obtain the partition function of the TQFT on a general (closed) manifold by cutting the manifold into small, elementary pieces which we understand, and then the partition function can be calculated from assembling the pieces together. This holds very generally in ...

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Basically the reason is that the classical conformal symmetry no longer holds at the quantum level due to the presence of the trace anomaly. More precisely, the tracelessness of the quantum stress-energy tensor is incompatible with the normal ordering needed to define it. By cohomological reasons, the trace of the stress-energy tensor, although ...

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Two dimensional CFTs separate into a left-moving sector and a right-moving sectors. The Virasoro generators $L_n$ act on the left-moving sector and ${\tilde L}_n$ act on the right-moving ones. Operators (or states due to the state-operator map) are labelled independently by representations of the left- and right-moving Virasoro. In particular, $h$ and ...

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There is a confusion and misunderstanding about what "perturbative" means and its relation with the power series expansion around certain small parameter. If you state that QED/QCD is non-perturbative, because amplitudes are defined exactly, and not as a power expansion, then the same applies to string theory. Amplitudes, in any QFT, is defined a sum over ...

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