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The normal ordering makes the expression of $b_{-r}\cdot b_{r+m}$ symmetric around the point $r=-\frac{m}{2}$, and these symmetric parts differ by a sign. This sign difference due to normal ordering makes $m/2$ redundancy.


Roughly, every chiral part of a rational CFT gives a TFT theory. For example for WZW models the chiral parts are current algebras. The corresponding TFT is Chern-Simons theory. The point is the representation of a chiral rational CFT is a modular tensor category. From a modular tensor category one can construct a 3D TFT via the Reshetikhin-Turaev ...


The Weyl transformation (unlike diffeomorphisms) does not affect physical fields, only the geometry of space-time. Probably the best way to show that your equation is Weyl-invariant is to build an action which (when varied) yields the equation. In your example it would be $$ S[\Psi] = \int d^4 x \: \sqrt{-g} \: \bar{\Psi} \gamma^a e^{\mu}_a D_{\mu} \Psi. $$ ...


A conformal transformation is one which alters the metric up to a factor, i.e. $$g_{\mu\nu}(x)\to\Omega^2(x)g_{\mu\nu}(x)$$ A field theory described by a Lagrangian invariant up to a total derivative under a conformal transformation is said to be a conformal field theory. These transformations include Scaling or dilations $x^\mu \to \lambda x^\mu$ ...


Imagine a QFT with some particle content. Some of these fields will be massless and some massive. For simplicity, consider a massles scalar field and a massive scalar field with mass $M$. If we are working at some energy $E\ll M$, we won't see the massive field (as happened with the Higgs before LHC, for example). This is the IR CFT. Why IR? Because we ...


I think that this nomenclature has nothing to do with the regulators. "UV CFT" and "IR CFT" actually refers to the end point of the RG flow which is triggered by a relevant operator that perturbs an UV fixed point, and it ends (barring certain non-unitary QFT) into a fixed point at lower energy scales, hence the name IR.

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