Network geometry with flavor: From complexity to quantum geometry

Here we introduce the Network Geometry with Flavor $s=-1,0,1$ (NGF) describing simplicial complexes defined in arbitrary dimension $d$ and evolving by a non-equilibrium dynamics. The NGF can generate discrete geometries of different nature, ranging from chains and higher dimensional manifolds to scale-free networks with small-world properties, scale-free degree distribution and non-trivial community structure. The NGF admits as limiting cases both the Bianconi-Barab\'asi model for complex networks the stochastic Apollonian network, and the recently introduced model for Complex Quantum Network Manifolds. The thermodynamic properties of NGF reveal that NGF obeys a generalized area law opening a new scenario for formulating its coarse-grained limit. The structure of NGF is strongly dependent on the dimensionality $d$. We also show that NGF admits a quantum mechanical description in terms of associated quantum network states. Quantum network states are evolving by a Markovian dynamics and a quantum network state at time $t$ encodes all possible NGF evolutions up to time $t$. Interestingly the NGF remains fully classical but its statistical properties reveal the relation to its quantum mechanical description. In fact the $\delta$-dimensional faces of the NGF have generalized degrees that follow either the Fermi-Dirac, Boltzmann or Bose-Einstein statistics depending on the flavor $s$ and the dimensions $d$ and $\delta$.Comment: (37 pages, 4 figures

Functional Multiplex PageRank

(7 pages, 5 figures)(7 pages, 5 figures)(7 pages, 5 figures

Further evidence for asymptotic safety of quantum gravity

The asymptotic safety conjecture is examined for quantum gravity in four dimensions. Using the renormalisation group, we find evidence for an interacting UV fixed point for polynomial actions up to the 34th power in the Ricci scalar. The extrapolation to infinite polynomial order is given, and the self-consistency of the fixed point is established using a bootstrap test. All details of our analysis are provided. We also clarify further aspects such as stability, convergence, the role of boundary conditions, and a partial degeneracy of eigenvalues. Within this setting we find strong support for the conjecture

Modified Dispersion Relations from the Renormalization Group of Gravity

We show that the running of gravitational couplings, together with a suitable identification of the renormalization group scale can give rise to modified dispersion relations for massive particles. This result seems to be compatible with both the frameworks of effective field theory with Lorentz invariance violation and deformed special relativity. The phenomenological consequences depend on which of the frameworks is assumed. We discuss the nature and strength of the available constraints for both cases and show that in the case of Lorentz invariance violation, the theory would be strongly constrained.Comment: revtex4, 9 pages, updated to match published versio

A nonsingular spacetime defect

A nonsingular localized static classical solution is constructed for standard Einstein gravity coupled to an $SO(3)\times SO(3)$ chiral model of scalars (Skyrme model). This solution corresponds to a spacetime defect and its construction proceeds in three steps. First, an \textit{Ansatz} is presented for a solution with nonsimply connected topology of the spacetime manifold. Second, an exact vacuum solution of the reduced field equations is obtained. Third, matter fields are included and a particular exact solution of the reduced field equations is found. The latter solution has a diverging total energy, but its existence at least demonstrates that a nonsingular defect-type solution having nonsimply connected topology is possible with nontrivial matter fields.Comment: 20 pages, v8: published versio

One Loop Beta Functions in Topologically Massive Gravity

We calculate the running of the three coupling constants in cosmological, topologically massive 3d gravity. We find that \nu, the dimensionless coefficient of the Chern-Simons term, has vanishing beta function. The flow of the cosmological constant and Newton's constant depends on \nu, and for any positive \nu there exist both a trivial and a nontrivial fixed point.Comment: 44 pages, 16 figure

Future of the universe in modified gravitational theories: Approaching to the finite-time future singularity

We investigate the future evolution of the dark energy universe in modified gravities including $F(R)$ gravity, string-inspired scalar-Gauss-Bonnet and modified Gauss-Bonnet ones, and ideal fluid with the inhomogeneous equation of state (EoS). Modified Friedmann-Robertson-Walker (FRW) dynamics for all these theories may be presented in universal form by using the effective ideal fluid with an inhomogeneous EoS without specifying its explicit form. We construct several examples of the modified gravity which produces accelerating cosmologies ending at the finite-time future singularity of all four known types by applying the reconstruction program. Some scenarios to resolve the finite-time future singularity are presented. Among these scenarios, the most natural one is related with additional modification of the gravitational action in the early universe. In addition, late-time cosmology in the non-minimal Maxwell-Einstein theory is considered. We investigate the forms of the non-minimal gravitational coupling which generates the finite-time future singularities and the general conditions for this coupling in order that the finite-time future singularities cannot emerge. Furthermore, it is shown that the non-minimal gravitational coupling can remove the finite-time future singularities or make the singularity stronger (or weaker) in modified gravity.Comment: 25 pages, no figure, title changed, accepted in JCA

Renormalization Group Flow in Scalar-Tensor Theories. II

We study the UV behaviour of actions including integer powers of scalar curvature and even powers of scalar fields with Functional Renormalization Group techniques. We find UV fixed points where the gravitational couplings have non-trivial values while the matter ones are Gaussian. We prove several properties of the linearized flow at such a fixed point in arbitrary dimensions in the one-loop approximation and find recursive relations among the critical exponents. We illustrate these results in explicit calculations in $d=4$ for actions including up to four powers of scalar curvature and two powers of the scalar field. In this setting we notice that the same recursive properties among the critical exponents, which were proven at one-loop order, still hold, in such a way that the UV critical surface is found to be five dimensional. We then search for the same type of fixed point in a scalar theory with minimal coupling to gravity in $d=4$ including up to eight powers of scalar curvature. Assuming that the recursive properties of the critical exponents still hold, one would conclude that the UV critical surface of these theories is five dimensional.Comment: 14 pages. v.2: Minor changes, some references adde

Quantum Einstein Gravity

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 particular 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 space-times 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.Comment: 87 pages, 13 figures, review article prepared for the New Journal of Physics focus issue on Quantum Einstein Gravit