The exact renormalization group equation for pure quantum gravity is used to derive the non-perturbative β-functions for the dimensionless Newton constant and cosmological constant on the theory space spanned by the Einstein-Hilbert truncation. The resulting coupled differential equations are evaluated for a sharp cutoff function. The features of these flow equations are compared to those found when using a smooth cutoff. The system of equations with sharp cutoff is then solved numerically, deriving the complete renormalization group flow of the Einstein-Hilbert truncation in d = 4. The resulting renormalization group trajectories are classified and their physical relevance is discussed. The non-trivial fixed point which, if present in the exact theory, might render Quantum Einstein Gravity nonperturbatively renormalizable is investigated for various spacetime dimensionalities.

We give a pedagogical introduction to the basic ideas and concepts of the Asymptotic Safety program in quantum Einstein gravity. Using the continuum approach based upon the effective average action, we summarize the state of the art of the field with a focus on the evidence supporting the existence of the non-trivial renormalization group fixed point at the heart of the construction. As an application, the multifractal structure of the emerging spacetimes is discussed in detail. In particular, we compare the continuum prediction for their spectral dimension with Monte Carlo data from the causal dynamical triangulation approach.

We construct two new versions of the c-map which allow us to obtain the target manifolds of hypermultiplets in Euclidean theories with rigid N = 2 supersymmetry. While the Minkowskian para-c-map is obtained by dimensional reduction of the Minkowskian vector multiplet lagrangian over time, the Euclidean para-c-map corresponds to the dimensional reduction of the Euclidean vector multiplet lagrangian. In both cases the resulting hypermultiplet target spaces are para-hyper-Kähler manifolds. We review and prove the relevant results of para-complex and para-hypercomplex geometry. In particular, we give a second, purely geometrical construction of both c-maps, by proving that the cotangent bundle N = T * M of any affine special (para-)Kähler manifold M is parahyper-Kähler.

Finding the exact, quantum corrected metric on the hypermultiplet moduli space in Type II string compactifications on Calabi-Yau threefolds is an outstanding open problem. We address this issue by relating the quaternionic-Kähler metric on the hypermultiplet moduli space to the complex contact geometry on its twistor space. In this framework, Euclidean D-brane instantons are captured by contact transformations between different patches. We derive those by recasting the previously known A-type D2-instanton corrections in the language of contact geometry, covariantizing the result under electro-magnetic duality, and using mirror symmetry. As a result, we are able to express the effects of all D-instantons in Type II compactifications concisely as a sum of dilogarithm functions. We conclude with some comments on the relation to microscopic degeneracies of four-dimensional BPS black holes and to the wall-crossing formula of Kontsevich and Soibelman, and on the form of the yet unknown NS5-brane instanton contributions.

We use the functional renormalization group equation for quantum gravity to construct a nonperturbative flow equation for modified gravity theories of the form S = d d x √ gf (R). Based on this equation we show that certain gravitational interactions monomials can be consistently decoupled from the renormalization group (RG) flow and reproduce recent results on the asymptotic safety conjecture. The non-perturbative RG flow of non-local extensions of the Einstein-Hilbert truncation including d d x √ g ln(R) and d d x √ gR −n interactions is investigated in detail. The inclusion of such interactions resolves the infrared singularities plaguing the RG trajectories with positive cosmological constant in previous truncations. In particular, in some R −n -truncations all physical trajectories emanate from a Non-Gaussian (UV) fixed point and are well-defined on all RG scales. The RG flow of the ln(R)-truncation contains an infrared attractor which drives a positive cosmological constant to zero dynamically.1

We study the non-perturbative renormalization group flow of higher-derivative gravity employing functional renormalization group techniques. The non-perturbative contributions to the β-functions shift the known perturbative ultraviolet fixed point into a non-trivial fixed point with three UV-attractive and one UV-repulsive eigendirections, consistent with the asymptotic safety conjecture of gravity. The implication of this transition on the unitarity problem, typically haunting higher-derivative gravity theories, is discussed.Among the many approaches to quantum gravity, a special place is occupied by higherderivative gravity, which, besides the Einstein-Hilbert term, also includes fourth-order operators in the action. Indeed, the higher-derivative propagators soften the divergences encountered in the perturbative quantization, rendering the theory perturbatively renormalizable [1] and asymptotically free at the one-loop level [2,3,4,5,6]. Unfortunately, the extra terms responsible for the improved UV behavior also induce massive negative norm states [7], so-called "poltergeists", which led to the belief that the theory is not unitary. Several arguments suggest that this shortcoming can be cured by quantum effects [2,8], but the lack of non-perturbative methods has made it hard to substantiate such claims.Recently, the question of renormalizability has received renewed attention due to mounting evidence in favor of the non-perturbative renormalizability, or asymptotic safety (AS), of gravity [9,10,11,12]. In this scenario, the ultraviolet (UV) behavior of the theory is controlled by a

Abstract:We extend the twistor methods developed in our earlier work on linear deformations of hyperkähler manifolds [1] to the case of quaternionic-Kähler manifolds. Via Swann's construction, deformations of a 4d-dimensional quaternionic-Kähler manifold M are in one-to-one correspondence with deformations of its 4d + 4-dimensional hyperkähler cone S. The latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space Z S , with a suitable homogeneity condition that ensures that the hyperkähler cone property is preserved. Equivalently, we show that the deformations of M can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space Z M of M, by-passing the Swann bundle and its twistor space. We specialize these general results to the case of quaternionic-Kähler metrics with d + 1 commuting isometries, obtainable by the Legendre transform method, and linear deformations thereof. We illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree-and one-loop level.

Many approaches to quantum gravity have resorted to diffusion processes to characterize the spectral properties of the resulting quantum spacetimes. We critically discuss these quantum-improved diffusion equations and point out that a crucial property, namely positivity of their solutions, is not preserved automatically. We then construct a novel set of diffusion equations with positive semidefinite probability densities, applicable to asymptotically safe gravity, Hořava-Lifshitz gravity and loop quantum gravity. These recover all previous results on the spectral dimension and shed further light on the structure of the quantum spacetimes by assessing the underlying stochastic processes. Pointing out that manifestly different diffusion processes lead to the same spectral dimension, we propose the probability distribution of the diffusion process as a refined probe of quantum spacetime.