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

Employing standard results from spectral geometry, we provide strong evidence that in the classical limit the ground state of three-dimensional causal dynamical triangulations is de Sitter spacetime. This result is obtained by measuring the expectation value of the spectral dimension on the ensemble of geometries defined by these models, and comparing its large scale behaviour to that of a sphere (Euclidean de Sitter). From the same measurement we are also able to confirm the phenomenon of dynamical dimensional reduction observed in this and other approaches to quantum gravity -the first time this has been done for three-dimensional causal dynamical triangulations. In this case, the value for the short-scale limit of the spectral dimension that we find is approximately 2. We comment on the relevance of these results for the comparison to asymptotic safety and Hořava-Lifshitz gravity, among other approaches to quantum gravity.

We show that in general a spacetime having a quantum group symmetry has also a scale dependent fractal dimension which deviates from its classical value at short scales, a phenomenon that resembles what observed in some approaches to quantum gravity. In particular we analyze the cases of a quantum sphere and of κ-Minkowski, the latter being relevant in the context of quantum gravity.In the quest for new physics at the Planck scale the idea that spacetime might become noncommutative [1,2,3] has gained a lot of attention, in particular for its potential phenomenological implications [4]. Whether such idea is supposed to be taken as a starting point for the construction a quantum theory of gravity (see for example [2]) or can be derived from it (for example [5]) there are several reasons why it could play a role at the Planck scale. Somehow postponing the issue of a more complete and fundamental theory most of the efforts in the literature have gone on the study of noncommutative versions of flat spacetime, which naively might be thought as a ground state of the full theory of quantum gravity.On the other hand constructive approaches to quantum gravity, such as causal dynamical triangulations (CDT) [6] and exact renormalization group (ERG) [7], which make no use of postulated new physics, have something interesting to say about Planck scale properties of spacetime. It is somehow surprising to see that apparently very different approaches give rise to very similar results as it is the case for the spectral dimension of spacetime: both in CDT [8] and in ERG [9] evidence has been given for the emergence of a (ground state) spacetime with fractal properties such as the effective (spectral) dimension d s varying from a classical value d s = 4 at large scales down to d s = 2 at short scales. It is a legitimate and interesting question to ask whether such a fractal nature of spacetime is compatible with the expectation of some sort of noncommutativity.An appealing realization of noncommutativity is that in which spacetime remains maximally symmetric but the Lie group of symmetries is deformed into a quantum group (as in [10]), a deformation also favoured by general arguments on the possible non-locality of a final quantum theory of gravity [11], and which constitutes a solid realization of the so-called Doubly Special Relativity [12,13]. Research in this area in still at an early stage and a complete formulation of quantum field theory based on a quantum group symmetry is still lacking, but some proposals have been put forward for the construction of the corresponding Fock space (see for example [14] and references therein). Here we explore the geometrical properties of such type of spacetimes by calculating the spectral dimension associated with them. In order to do * Electronic address: dbenedetti@perimeterinstitute.ca so we adopt a group theoretical construction that suits well to the quantum group formalism. We find for the noncommutative spacetimes considered a result qualitatively similar to that found in CDT and ER...

Within the context of the functional renormalization group flow of gravity, we suggest that a generic f(R) ansatz (i.e. not truncated to any specific form, polynomial or not) for the effective action plays a role analogous to the local potential approximation (LPA) in scalar field theory. In the same spirit of the LPA, we derive and study an ordinary differential equation for f(R) to be satisfied by a fixed point of the renormalization group flow. As a first step in trying to assess the existence of global solutions (i.e. true fixed point) for such equation, we investigate here the properties of its solutions by a comparison of various series expansions and numerical integrations. In particular, we study the analyticity conditions required because of the presence of fixed singularities in the equation, and we develop an expansion of the solutions for large R up to order N=29. Studying the convergence of the fixed points of the truncated solutions with respect to N, we find a characteristic pattern for the location of the fixed points in the complex plane, with one point stemming out for its stability. Finally, we establish that if a non-Gaussian fixed point exists within the full f(R) approximation, it corresponds to an R^2 theory

Abstract:We set up the Functional Renormalisation Group formalism for Tensorial Group Field Theory in full generality. We then apply it to a rank-3 model over U(1) 3 , endowed with a kinetic term linear in the momenta and with nonlocal interactions. The system of FRG equations turns out to be non-autonomous in the RG flow parameter. This feature is explained by the existence of a hidden scale, the radius of the group manifold. We investigate in detail the opposite regimes of large cut-off (UV) and small cut-off (IR) of the FRG equations, where the system becomes autonomous, and we find, in both case, Gaussian and non-Gaussian fixed points. We derive and interpret the critical exponents and flow diagrams associated with these fixed points, and discuss how the UV and IR regimes are matched. Finally, we discuss the evidence for a phase transition from a symmetric phase to a broken or condensed phase, from an RG perspective, finding that this seems to exist only in the approximate regime of very large radius of the group manifold, as to be expected for systems on compact manifolds.