## Quantum Gravity and the Functional Renormalization Group: The Road towards Asymptotic SafetyDuring the past two decades the gravitational asymptotic safety scenario has undergone a major transition from an exotic possibility to a serious contender for a realistic theory of quantum gravity. It aims at a mathematically consistent quantum description of the gravitational interaction and the geometry of spacetime within the realm of quantum field theory, which keeps its predictive power at the highest energies. This volume provides a self-contained pedagogical introduction to asymptotic safety, and introduces the functional renormalization group techniques used in its investigation, along with the requisite computational techniques. The foundational chapters are followed by an accessible summary of the results obtained so far. It is the first detailed exposition of asymptotic safety, providing a unique introduction to quantum gravity and it assumes no previous familiarity with the renormalization group. It serves as an important resource for both practising researchers and graduate students entering this maturing field. |

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anomalous dimension ansatz approximation Asymptotic Safety background field Background Independence background metric beta functions bi-metric BRST classical coefficients compute coordinate corresponding cosmological constant coupling constants covariant derivatives CREH critical exponents cutoff defined denotes dependence diffeomorphism differential equation dimensional dimensionless dynamical Effective Average Action eigenvalues Einstein Einstein–Hilbert truncation evaluated example expansion expectation value finite fixed point flow equation fluctuation FRGE functional integral gauge invariance gauge transformations gauge-fixing ghost given gºv heat kernel hºw linear manifold matrix matter fields modes Newton constant NGFP non-perturbative obtained operator parameter perturbative Phys physical potential properties quantization quantum field theory quantum gravity regime renormalization group respectively result RG equation RG flow RG trajectories scalar scale scale-dependent shape function ſhep-th single-metric ſº solution spacetime structure tensor theory space threshold functions tion Tºg vanishes variables vector field Ward identity