We discuss the effective potential of the conformal factor in the effective average action approach to Quantum Einstein Gravity. Without invoking any truncation or other approximations we show that if the theory has has a non-Gaussian ultraviolet fixed point and is asymptotically safe the potential has a characteristic behavior near the origin. We argue that this characteristic behavior has already been observed in numerical simulations within the Causal Dynamical Triangulation approach.
We review some of our recent work on the conformal geometry corresponding to the triangulated surfaces used in 2-dimensional simplicial quantum gravity. In particular, we discuss the regularized Liouville action associated with random Regge triangulations and its connection with Hodge-Deligne theory.
In the dynamical triangulation model of four dimensional euclidean quantum gravity we investigate gravitational binding. Two scalar test particles (quenched approximation) have a positive binding energy, thereby showing that the model can represent gravitational attraction.
We formulate two-dimensional rational conformal field theory as a natural generalization of two-dimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A in the modular category that encodes the Moore-Seiberg data of the underlying chiral CFT. Just like for lattice TFTs, this algebra is itself not an observable quantity. Rather, Morita equivalent algebras give rise to equivalent theories. Morita equivalence also allows for a simple understanding of …
We construct a function of the edge-lengths of a triangulated surface whose variation under a rescaling of all the edges that meet at a vertex is the defect angle at that vertex. We interpret this function as a gravitational effective action on the triangulation, and the variation as a trace anomaly.
We recall how the Gauss-Bonnet theorem can be interpreted as a finite dimen- sional index theorem. We describe the construction given in hep-th/0512293 of a function that can be interpreted as a gravitational effective action on a triangulation. The variation of this function under local rescalings of the edge lengths sharing a vertex is the Euler density, and we use it to illustrate how continuous concepts can have natural discrete analogs.
We show how Boundary Conformal Field Theory deformation techniques allow for a complete characterisation of the coupling between the discrete geometry inherited uniformizing a random Regge triangulations and open string theory.
The four dimensional Causal Dynamical Triangulations (CDT) approach to quantum gravity is already more than ten years old theory with numerous unprecedented predictions such as non-trivial phase structure of gravitational field and dimensional running. Here, we discuss possible empirical consequences of CDT derived based on the two features of the approach mentioned above. A possibility of using both astrophysical and cosmological observations to test CDT is discussed. We show that scenarios which can be ruled out at the empirical level exist.
Our goal here is to present a detailed analysis connecting the anomalous scaling properties of 2D simplicial quantum gravity to the geometry of the moduli space M_{g,N} of genus g Riemann surfaces with N punctures. In the case of pure gravity we prove that the scaling properties of the set of dynamical triangulations with N vertices are directly provided by the large N asymptotics of the Weil-Petersson volume of M_{g,N}, recently discussed by Manin and Zograf. Such a geometrical characterization explains why dynamical triangulations automatically take into account the anomalous scaling propert…
The Causal Dynamical Triangulation (CDT) approach to quantum gravity is a lattice approximation to the gravitational path integral. Developed by Ambjørn, Jurkiewicz and Loll, it has yielded some important results, notably the emergence of classical spacetime and short scale dimensional reduction. However, virtually all the results reported so far have been based on a single computer code. In this paper we present the first completely independent verification of the CDT algorithm, and report the successful reproduction of the emergence of classical spacetime and smooth reduction in the spectral…
This short note is devoted to the Hamiltonian formulation of the conformal decomposition of the gravitational field that was performed in [gr-qc/0501092]. We also analyze the gauge fixed form of the theory when we fix the conformal symmetry by imposing the condition det g=1.
We investigate the evolution of cosmological perturbations generated during de Sitter inflation in the conformal gravity. Primordial gravitational waves are composed of vector and tensor modes. We obtain the constant vector and tensor power spectra which seems to be correct because the conformal gravity is invariant under conformal transformation like the Maxwell kinetic term.
We propose a method to holographically compute the conformal partial waves in any decomposition of correlation functions of primary operators in conformal field theories using open Wilson network operators in the holographic gravitational dual. The Wilson operators are the gravitational ones where gravity is written as a gauge theory in the first order Hilbert-Palatini formalism. We apply this method to compute the global conformal blocks and partial waves in 2d CFTs reproducing many of the known results.
Hairy black holes in the gravitational decoupling setup are studied from the perspective of conformal anomalies. Fluctuations of decoupled sources can be computed by measuring the way the trace anomaly-to-holographic Weyl anomaly ratio differs from unit. Therefore the gravitational decoupling parameter governing three hairy black hole metrics is then bounded to a range wherein one can reliably emulate AdS/CFT with gravitational decoupled solutions, in the tensor vacuum regime.
We study a lattice regularization of the gravitational path integral--causal dynamical triangulations--for (2+1)-dimensional Einstein gravity with positive cosmological constant in the presence of past and future spacelike boundaries of fixed intrinsic geometries. For spatial topology of a 2-sphere, we determine the form of the Einstein-Hilbert action supplemented by the Gibbons-Hawking-York boundary terms within the Regge calculus of causal triangulations. Employing this action we numerically simulate a variety of transition amplitudes from the past boundary to the future boundary. To the ext…
We discuss in some detail the interaction of classical particles, including the scattering and memory effect, with a pulse of gravitational plane wave. The key point is the conformal symmetry of gravitational plane waves. In particular, we obtain, in the limit of short pulse, some results for impulsive gravitational waves. Furthermore, in the general case, we give certain conditions which allow us to completely describe the interaction in terms of the singular Baldwin-Jeffery-Rosen coordinates.
A novel continuum theory of two-dimensional quantum gravity, based on a version of Causal Dynamical Triangulations which incorporates topology change, has recently been formulated as a genuine string field theory in zero-dimensional target space (arXiv:0802.0719). Here we show that the Dyson-Schwinger equations of this string field theory are reproduced by a cubic matrix model. This matrix model also appears in the so-called Dijkgraaf-Vafa correspondence if the superpotential there is required to be renormalizable. In the spirit of this model, as well as the original large-N expansion by 't Ho…
We take a step toward a nonperturbative gravitational path integral for black-hole geometries by deriving an expression for the expansion rate of null geodesic congruences in the approach of causal dynamical triangulations. We propose to use the integrated expansion rate in building a quantum horizon finder in the sum over spacetime geometries. It takes the form of a counting formula for various types of discrete building blocks which differ in how they focus and defocus light rays. In the course of the derivation, we introduce the concept of a Lorentzian dynamical triangulation of product typ…
We extend the discrete Regge action of causal dynamical triangulations to include discrete versions of the curvature squared terms appearing in the continuum action of (2+1)-dimensional projectable Horava-Lifshitz gravity. Focusing on an ensemble of spacetimes whose spacelike hypersurfaces are 2-spheres, we employ Markov chain Monte Carlo simulations to study the path integral defined by this extended discrete action. We demonstrate the existence of known and novel macroscopic phases of spacetime geometry, and we present preliminary evidence for the consistency of these phases with solutions t…
"Causal Dynamical Triangulations" (CDT) represent a lattice regularization of the sum over spacetime histories, providing us with a non-perturbative formulation of quantum gravity. The ultraviolet fixed points of the lattice theory can be used to define a continuum quantum field theory, potentially making contact with quantum gravity defined via asymptotic safety. We describe the formalism of CDT, its phase diagram, and the quantum geometries emerging from it. We also argue that the formalism should be able to describe a more general class of quantum-gravitational models of Horava-Lifshitz typ…
We investigate numerically 10 - dimensional Euclidean quantum gravity (with discretized Einstein - Hilbert action) in the framework of the dynamical triangulation approach. For the considered values of the gravitational coupling we observed two phases, the behavior of which is found to be similar to that of the crumpled and elongated phases of 3, 4 and 5 dimensional models. Surprisingly, (for the observed lattice sizes) the natural state of the 10 D system (when the Einstein - Hilbert action is turned off) is found to resemble branched polymer while in the low dimensional systems the natural s…
We present a classical conformal field theory on an arbitrary two-dimensional spacetime background. The dynamical object is a space-filling string, and the evolution may be thought as occurring on the manifold of the conformal group. The theory is a "descendant" of the theory of gravitation in two-dimensional spacetime. The discussion is based on the relation of the deformations of the space-filling string with conformal transformations. The realization of the conformal algebra in terms of surface deformations possesses a classical central charge. The action principle, the conformal and Weyl i…
It is believed that in the near future, gravitational wave detections will become a promising tool not only to test gravity theories, but also to probe extremely curved spacetime regions in our universe, such as the surroundings of black holes. In this paper, we investigate the quasinormal modes (QNMs) of the axial gravitational perturbations of a class of non-singular black holes conformally related to the Schwarzschild black hole. These non-singular black holes can be regarded as the vacuum solution of a family of conformal gravity theories which are invariant under conformal transformations…
We consider a Georgi-Glashow model conformally coupled to gravity. The conformally invariant action includes a triplet of scalar fields and SO(3) non-Abelian gauge fields. However, the usual mass term $μ^2 ϕ^2$ is forbidden by the symmetry and this role is now played by the conformal coupling of the Ricci scalar to the scalar fields. Spontaneous symmetry breaking occurs via gravitation. The spherically symmetric solutions correspond to localized solitons (magnetic monopoles) in asymptotically anti-de-Sitter (AdS) space-time and the metric outside the core of the monopole is found to be Schwarz…
We consider the gravitational radiation in conformal gravity theory. We perturb the metric from flat Mikowski space and obtain the wave equation after introducing the appropriate transformation for perturbation. We derive the effective energy-momentum tensor for the gravitational radiation, which can be used to determine the energy carried by gravitational waves.
We derive the leading quantum corrections to the gravitational potentials in a de Sitter background, due to the vacuum polarization from loops of conformal fields. Our results are valid for arbitrary conformal theories, even strongly interacting ones, and are expressed using the coefficients $b$ and $b'$ appearing in the trace anomaly. Apart from the de Sitter generalization of the known flat-space results, we find two additional contributions: one which depends on the finite coefficients of terms quadratic in the curvature appearing in the renormalized effective action, and one which grows lo…
We study non-Einstein Bach-flat gravitational instanton solutions that can be regarded as the generalization of the Taub-NUT/Bolt and Eguchi-Hanson solutions of Einstein gravity to conformal gravity. These solutions include non-Einstein spaces which are either asymptotically locally flat spacetimes (ALF) or asymptotically locally Anti-de Sitter (AlAdS). Nevertheless, solutions with different asymptotic conditions exist: we find geometries that present a weakened AlAdS asymptotia, exhibiting the typical low decaying mode of conformal gravity. This permits to identify the simple Neumann boundary…
Gravitational currents and charges from conformal Killing-Yano tensors
The Blackett Laboratory, Imperial College London（帝国理工学院布莱克特实验室）；Department of Physics and Astronomy, Uppsala University（乌普萨拉大学物理与天文学系）；Centre for Geometry and Physics, Uppsala University（乌普萨拉大学几何与物理中心）
We construct isometric and conformally isometric embeddings of some gravitational instantons in $\mathbb{R}^8$ and $\mathbb{R}^7$. In particular we show that the embedding class of the Einstein--Maxwell instanton due to Burns is equal to $3$. For $\mathbb{CP}^2$, Eguchi--Hanson and anti-self-dual Taub-NUT we obtain upper and lower bounds on the embedding class.
Unification of Conformal and Fuzzy Gravities with Internal Interactions - study of their behaviour in low energies and possible signals in the detection of Gravitational Waves
Thermal partition functions for gravitational systems have traditionally been studied using Euclidean path integrals. But in Euclidean signature the gravitational action suffers from the conformal factor problem, which renders the action unbounded below. This makes it difficult to take the Euclidean formulation as fundamental. However, despite their familiar association with periodic imaginary time, thermal gravitational partition functions can also be described by real-time path integrals over contours defined by real Lorentzian metrics. The one caveat is that we should allow certain codimens…
Could a conformal Galileon be describing a gauge mode of a broader diffeomorphism invariant theory? We answer this question affirmatively in 3D by using a coset construction for a nonlinearly realized conformal symmetry. In particular, we show that the conformal Galileon emerges as a Stückelberg field of a local Weyl symmetry. The coset construction gives us a diffeomorphism and Weyl invariant 3D theory containing first, second, and third powers of the curvatures, their couplings to certain tensors made of the Galileon and its derivatives, and the conformal Galileon terms. In a theory with a b…
We discuss the geometry of dynamical triangulations associated with 3-dimensional and 4-dimensional simplicial quantum gravity. We provide analytical expressions for the canonical partition function in both cases, and study its large volume behavior. In the space of the coupling constants of the theory, we characterize the infinite volume line and the associated critical points. The results of this analysis are found to be in excellent agreement with the MonteCarlo simulations of simplicial quantum gravity. In particular, we provide an analytical proof that simply-connected dynamically triangu…
Carroll-Cotton Tensors and Gravitational Radiation at Null Infinity
École polytechnique, Centre National de la Recherche Scientifique – Unité Mixte de Recherche 7644, Institut Polytechnique de Paris（巴黎综合理工学院，法国国家科学研究中心联合研究单位7644，巴黎理工学院）；Université Libre de Bruxelles and International Solvay Institutes（布鲁塞尔自由大学和国际索尔维研究所）；International School for Advanced Studies (SISSA)（高级国际研究学院（SISSA））；Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；Université de Mons（蒙斯大学）
We study conformal gravity as an alternative theory of gravitation. For conformal gravity to be phenomenologically viable requires that the conformal symmetry is not manifest at the energy scales of the other known physical forces. Hence we require a mechanism for the spontaneous breaking of conformal invariance. In this paper we study the possibility that conformal invariance is spontaneously broken due to interactions with conformally coupled matter fields. The vacuum of the theory admits conformally non-invariant solutions corresponding to maximally symmetric space-times and variants thereo…
A hybrid model which allows to interpolate between the (original) Regge approach and dynamical triangulations is introduced. The gained flexibility in the measure is exploited to study dynamical triangulation in a fixed geometry. Our numerical results support KPZ exponents. A critical assessment concerning the apparent lack of gravitational effects in two dimensions follows.
It has been conjectured that a near-extreme Kerr black hole is described by a 2d CFT. Previous work has shown that CFT operators dual to axisymmetric gravitational perturbations have integer conformal weights. In this paper, we study the analogous problem in 5d. We consider the most general near-extreme vacuum black hole with two rotational symmetries. This includes Myers-Perry black holes, black rings and Kaluza-Klein black holes. We find that operators dual to gravitational (or electromagnetic or massless scalar field) perturbations preserving both rotational symmetries have integer conforma…
We study spherically-symmetric structures in Conformal Gravity and in a scalar-tensor extension and gain some more insight about these gravitational theories. In both cases we analyze solutions in two systems: perfect fluid solutions and boson stars of a self-interacting complex scalar field. In the purely tensorial (original) theory we find in a certain domain of parameter space finite mass solutions with a linear gravitational potential but without a Newtonian contribution. The scalar-tensor theory exhibits a very rich structure of solutions whose main properties are discussed. Among them, s…
We demonstrate that relativistic conformal hydrodynamics in 2+1 dimensions displays a turbulent behaviour which cascades energy to longer wavelengths on both flat and spherical manifolds. Our motivation for this study is to understand the implications for gravitational solutions through the AdS/CFT correspondence. The observed behaviour implies gravitational perturbations of the corresponding black brane/black hole spacetimes (for sufficiently large scales/temperatures) will display a similar cascade towards longer wavelengths.
We compute in linearized gravity all the contributions to the gravitational effective action due to a virtual Dirac fermion, related to the conformal anomaly. This requires, in perturbation theory, the identification of the gauge-gauge-graviton vertex off mass shell, involving the correlator of the energy-momentum tensor and two vector currents ($TJJ$), which is responsible for the generation of the gauge contributions to the conformal anomaly in gravity. We also present the anomalous effective action in the inverse mass of the fermion as in the Euler-Heisenberg case.
In recent years, the use of conformal transformation techniques has become widespread in the literature on gravitational theories alternative to general relativity, on cosmology, and on nonminimally coupled scalar fields. Typically, the transformation to the Einstein frame is generated by a fundamental scalar field already present in the theory. In this context, the problem of which conformal frame is the physical one has to be dealt with and, in the general case, it has been clarified only recently; the formulation of a theory in the ``new'' conformal frame leads to departures from canonical …
Starting from a covariant and background independent definition of normal ordered vertex operators we give an alternative derivation of the KPZ relation between conformal dimensions and their gravitational dressed partners. With our method we are able to study for arbitrary genus the dependence of N-point functions on all dimensionful parameters. Implications for the interpretation of gravitational dressed dimensions are discussed.
The universal behavior of Hawking radiation is originated in the conformal symmetries of matter fields near the black hole horizon. We explain the origin of this universality based on (1) the gravitational anomaly and its higher-spin generalizations and (2) conformal transformation properties of fluxes.
A conformal field theory on the boundary of three-dimensional asymptotic anti-de Sitter spaces which appear as near horizon geometry of D-brane bound states is discussed. It is shown that partition functions of gravitational instantons appear as high and low temperature limits of the partition function of the conformal field theory. The result reproduces phase transition between the anti-de Sitter space and the BTZ black hole in the bulk gravity.
We present a model in which the breackdown of conformal symmetry of a quantum stress-tensor due to the trace anomaly is related to a cosmological effect in a gravitational model. This is done by characterizing the traceless part of the quantum stress-tensor in terms of the stress-tensor of a conformal invariant classical scalar field. We introduce a conformal frame in which the anomalous trace is identified with a cosmological constant. In this conformal frame we establish the Einstein field equations by connecting the quantum stress-tensor with the large scale distribution of matter in the un…
Using the non-symmetric-connection approach proposed by Osborn, we demonstrate that, for a bosonic string in a specially chosen plane-fronted gravitational wave and an axion background, the conformal anomaly vanishes at the two-loop level. Under some conditions, the anomaly vanishes at all orders.
We consider spinor representations of the conformal group. The spacetime is constructed by the 15-dimensional vectors in the adjoint representation of $SO(2,4)$. On the spacetime, we construct a gravitational model that is invariant under local transformation.
We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the $SO(3)$--invariant gravitational instantons. On a hyper--Kähler four--manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self--dual magnetic field. In the case of the anti--self--dual Taub NUT instanton we integrate these equations completely by separating the Hamilton--Jacobi equations, and finding a commuting set of first integrals. This gives the first example of an integrable conformal geodesic flow on a four--manifold whi…
We study a gravitational model whose vacuum sector is invariant under conformal transformations. In this model we investigate the anomalous gravitational coupling of the large-scale matter. In this kind of coupling the large-sale matter is taken to couple to a metric which is different but conformally related to the metric appearing explicitly in the vacuum sector. The effect of the conformal symmetry breaking of the large-scale matter would lead in general to a variable strength of the anomalous gravitational coupling. This feature is used to derive a relativistic particle concept which share…
We describe the motivation behind the recent formulation of a nonperturbative path integral for Lorentzian quantum gravity defined through Causal Dynamical Triangulations (CDT). In the case of two dimensions the model is analytically solvable, leading to a genuine continuum theory of quantum gravity whose ground state describes a two-dimensional "universe" completely governed by quantum fluctuations. One observes that two-dimensional Lorentzian and Euclidean quantum gravity are distinct. In the second part of the review we address the question of how to incorporate a sum over space-time topolo…
We show that a previously proposed new mechanism to eliminate quadratic divergences for scalar masses is self-consistently compatible with corrections induced by perturbative quantum gravity, provided the theory embeds consistently into a UV completion at the Planck scale.
We construct the three point function involving an axial vector current and two energy-momentum tensors for four dimensional conformal field theories. Conformal symmetry determines the form of this three point function uniquely up to a constant factor if the necessary conservation conditions are imposed. The gravitational axial anomaly present on a curved space background leads to a non-zero contribution for the divergence of the axial current in this three point function even on flat space. Using techniques related to differential regularisation which guarantee that the energy-momentum tensor…
A Higgsless model for strong, electro-weak and gravitational interactions is proposed. This model is based on the local symmetry group SU(3)xSU(2)xU(1)xC where C is the local conformal symmetry group. The natural minimal conformally invariant form of total lagrangian is postulated. It contains all Standard Model fields and gravitational interaction. Using the unitary gauge and the conformal scale fixing conditions we can eliminate all four real components of the Higgs doublet in this model. However the masses of vector mesons, leptons and quarks are automatically generated and are given by the…
