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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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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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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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