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We explore the nonperturbative renormalization group flow of Quantum EinsteinGravity (QEG) on an infinite dimensional theory space. We consider "conformallyreduced" gravity where only fluctuations of the conformal factor are quantizedand employ the Local Potential Approximation for its effective average action.The requirement of "background independence" in quantum gravity entails apartial differential equation governing the scale dependence of the potentialfor the conformal factor which differs significantly from that of a scalarmatter field. In the infinite dimensional space of potential functions we finda Gaussian as well as a non-Gaussian fixed point which provides furtherevidence for the viability of the asymptotic safety scenario. The analog of theinvariant cubic in the curvature which spoils perturbative renormalizability isseen to be unproblematic for the asymptotic safety of the conformally reducedtheory. The scaling fields and dimensions of both fixed points are obtainedexplicitly and possible implications for the predictivity of the theory arediscussed. Spacetime manifolds with $R^d$ as well as $S^d$ topology areconsidered. Solving the flow equation for the potential numerically we obtainexamples of renormalization group trajectories inside the ultraviolet criticalsurface of the non-Gaussian fixed point. The quantum theories based upon someof them show a phase transition from the familiar (low energy) phase of gravitywith spontaneously broken diffeomorphism invariance to a new phase of unbrokendiffeomorphism invariance; the latter phase is characterized by a vanishingexpectation value of the metric.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Conformal sector of Quantum Einstein Gravity in the local potential  approximation: non-Gaussian fixed point and a phase of unbroken  diffeomorphism invariance