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Background-free quantum gravity based on conformal gravity and conformal field theory on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>M</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:math>
We study four-dimensional quantum gravity formulated as a certain conformal field theory at the ultraviolet fixed point, whose dynamics is described by the combined system of Riegert-Wess-Zumino and Weyl actions. Background-free nature comes out as quantum diffeomorphism symmetry by quantizing the conformal factor of the metric field nonperturbatively. In this …