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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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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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