Abstract
The asymptotic safety program builds on a high-energy completion of gravity based on the Reuter fixed point, a non-trivial fixed point of the gravitational renormalization group flow. At this fixed point the canonical mass-dimension of coupling constants is balanced by anomalous dimensions induced by quantum fluctuations such that the theory enjoys quantum scale invariance in the ultraviolet. The crucial role played by the quantum fluctuations suggests that the geometry associated with the fixed point exhibits non-manifold like properties. In this work, we continue the characterization of this geometry employing the composite operator formalism based on the effective average action. Explicitly, we give a relation between the anomalous dimensions of geometric operators on a background d-sphere and the stability matrix encoding the linearized renormalization group flow in the vicinity of the fixed point. The eigenvalue spectrum of the stability matrix is analyzed in detail and we identify a perturbative regime where the spectral properties are governed by canonical power counting. Our results recover the feature that quantum gravity fluctuations turn the (classically marginal) R^2-operator into a relevant one. Moreover, we find strong indications that higher-order curvature terms present in the two-point function play a crucial role in guaranteeing the predictive power of the Reuter fixed point.
Highlights
General relativity taught us to think of gravity in terms of geometric properties of spacetime
The technical details underlying our computation have been relegated to three appendices: Appendix A reviews the technical background for evaluating operator traces using the early-time expansion of the heatkernel, Appendix B derives the beta functions governing the renormalization group flow of gravity in the Einstein-Hilbert truncation employing geometric gauge [61, 62], and Appendix C lists the two-point functions entering into the computation
In order to characterize the quantum geometry related to Asymptotic Safety, we study the properties of this matrix at the Reuter fixed point found in Appendix B [cf
Summary
General relativity taught us to think of gravity in terms of geometric properties of spacetime. Within the asymptotic safety program [22, 23], reviewed in Percacci [24], Litim [25], Reuter and Saueressig [26], Ashtekar et al [27], and Eichhorn [28], these quantities have been studied based on the composite operator formalism [19, 29,30,31,32] This formalism allows to determine the anomalous scaling dimension of geometric operators based on an approximation of the quantum-corrected graviton propagator. The purpose of present work is two-fold: Firstly, we extend the analysis [32] beyond the single-operator approximation and compute the complete matrix of anomalous dimensions associated with the class (1) This information allows to access the spectrum of the scaling matrix. The technical details underlying our computation have been relegated to three appendices: Appendix A reviews the technical background for evaluating operator traces using the early-time expansion of the heatkernel, Appendix B derives the beta functions governing the renormalization group flow of gravity in the Einstein-Hilbert truncation employing geometric gauge [61, 62], and Appendix C lists the two-point functions entering into the computation
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