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A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In 2D, the infinite-dimensional algebra of local conformal transformations normally permits exact solution or classification of such theories. Further use for CFT applications to string theory, statistical mechanics, and condensed matter physics.

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Question regarding the metric tensor and conformal transformations

OP provided this as an example where they read the equation in question. There the flat metric $\eta_{\mu\nu}$ is used (at least in the beginning). In equation 1.1' it is explicitely written that $ds' …
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