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Firstly, I'd like to recommend Conformal Field Theory by Di-Francesco, it is a comprehensive text which is thorough and contains many applications of conformal field theory. The text is indispensable. In conformal field theory, it is often characteristic of correlation functions to diverge as points of two or more fields coincide. The operator product ...

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The two-dimensional Polyakov action for a string with worldsheet $\Sigma$ and worldsheet metric $h_{ab}$ $$\frac{T}{2}\int_\Sigma \sqrt{-h}h^{ab}g_{\mu\nu}\partial_aX^\mu\partial_bX^\nu$$ has full conformal symmetry under the Virasoro algebra and under Weyl transformations1 , which can be seen as gauge degrees of freedom. It follows that we can always ...

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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.

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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 ...

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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 $$S[\Psi] = \int d^4 x \: \sqrt{-g} \: \bar{\Psi} \gamma^a e^{\mu}_a D_{\mu} \Psi.$$ ...

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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 $x^\mu \to \lambda x^\mu$ ...

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