Transition Amplitudes in 3D Quantum Gravity: Boundaries and Holography in the Coloured Boulatov Model

We consider transition amplitudes in the coloured simplicial Boulatov model for three-dimensional Riemannian quantum gravity. First, we discuss aspects of the topology of coloured graphs with non-empty boundaries. Using a modification of the standard rooting procedure of coloured tensor models, we then write transition amplitudes systematically as topological expansions. We analyse the transition amplitudes for the simplest boundary topology, the 2-sphere, and prove that they factorize into a sum entirely given by the combinatorics of the boundary spin network state and that the leading order is given by graphs representing the closed 3-ball in the large N limit. This is the first step towards a more detailed study of the holographic nature of coloured Boulatov-type GFT models for topological field theories and quantum gravity.Comment: 42+15 pages, 28+14 figures; revised version matching article published in Annales Henri Poincar\'

Gravitation Quantique 3D Quasi-Local : Amplitude Exact et Holographie

This thesis is dedicated to the study of quasi-local boundary in quantum gravity. In particular, we focus on the three-dimensional space-time case. This research takes root in the holographic principle, which conjectures that the geometry and the dynamic of a space-time region can be entirely described by a theory living on the boundary of this given region. The most studied case of this principle is the AdS/CFT correspondence, where the quantum fluctuations of an asymptotically AdS space are described by a conformal field theory living at spatial infinity, invariant under the Virasoro group. The philosophy applied in this thesis however differs from the AdS/CFT case. I chose to focus on the case of quasi-local holography, i.e. for a bounded region of space-time with a boundary at a finite distance. The objective is to clarify the bulk-boundary relation in quantum gravity described by the Ponzano-Regge model, which defined a model for 3D gravity via a discrete path integral.I present the first perturbative and exact computations of the Ponzano-Regge amplitude on a torus with a 2D boundary state. After the presentation of the general framework for the 3D amplitude in terms of the 2D boundary state, we focus on the case of the 2D torus, that found an application in the study of the thermodynamics of the BTZ black hole. First, the 2D boundary is described by a coherent spin network state in the semi-classical regime. The stationary phase approximation allows to recover in the asymptotic limit the usual amplitude for 3D quantum gravity as the character of the symmetry of asymptotically flat gravity, the BMS group. Then we introduce a new type of coherent boundary state, which allows an exact evaluation of the amplitude for 3D quantum gravity. We obtain a complex regularization of the BMS character. The possibility of this exact computation suggests the existence of a (quasi)-integrable structure, linked to the symmetries of 3D quantum gravity with 2D finite boundary.Cette thèse est consacrée à l’étude du rôle des frontières en gravitation quantique pour une région compacte de l’espace-temps et explore en détail le cas en trois dimensions d'espace-temps. Cette étude s'inscrit dans le contexte du principe holographique qui conjecture que la géométrie d'une région de l’espace et sa dynamique peuvent être entièrement décrites par une théorie vivant sur la frontière de cette région. La réalisation la plus étudiée de ce principe est la correspondance AdS/CFT, où les fluctuations quantiques d'une géométrie asymptotiquement AdS sont décrites par une théorie conforme sur la frontière à l'infini, invariante sous le groupe de Virasoro. La philosophie appliquée ici diffère d’AdS/CFT. Je me suis intéressé à une holographie quasi- locale, c’est-à-dire pour une région bornée de l’espace avec une frontière à distance finie. L'objectif est de clarifier la relation bulk-boundary dans le cadre du modèle de Ponzano-Regge, qui définit la gravitation quantique euclidienne en 3D par une intégrale de chemin discrète.Je présente les premiers calculs approximatifs et exacts des amplitudes de Ponzano-Regge avec un état quantique de frontière 2D. Après présentation générale du calcul de l'amplitude 3d en fonction de l'état quantique de bord 2D, on se concentre sur le cas d'un tore 2D, qui trouve application dans l'étude de la thermodynamique des trous noirs BTZ. Dans un premier temps, la frontière 2D est décrite par des états de spin networks semi-classiques. L'approximation par phase stationnaire permet de retrouver dans la limite asymptotique la formule de l'amplitude de la gravité quantique 3D en tant que caractère du groupe BMS des symétries d'un espace-temps asymptotiquement plat. Puis dans un second temps, on introduit de nouveaux états quantiques cohérents, qui permettent une évaluation analytique exacte des amplitudes de gravité quantique 3D à distance finie sous la forme d'une régularisation complexe du caractère BMS. La possibilité de ce calcul exact suggère l’existence de structures (quasi-) intégrables liées aux symétries de la gravité quantique 3D en présence de frontières 2D bornées

Dual diffeomorphisms and finite distance asymptotic symmetries in 3d gravity

We study the finite distance boundary symmetry current algebra of the most general first order theory of 3d gravity. We show that the space of quadratic generators contains diffeomorphisms but also a notion of dual diffeomorphisms, which together form either a double Witt or centreless BMS$_3$ algebra. The relationship with the usual asymptotic symmetry algebra relies on a duality between the null and angular directions, which is possible thanks to the existence of the dual diffeomorphisms

Probing the Shape of Quantum Surfaces: the Quadrupole Moment Operator

International audienceThe standard toolkit of operators to probe quanta of geometry in loop quantum gravity consists in area and volume operators as well as holonomy operators. New operators have been defined, in the framework for intertwiners, which allow one to explore the finer structure of quanta of geometry. However these operators do not carry information on the global shape of the intertwiners. Here we introduce dual multipole moments for continuous and discrete surfaces, defined through the normal vector to the surface, taking special care to maintain parametrization invariance. These are raised to multipole operators probing the shape of quantum surfaces. Further focusing on the quadrupole moment, we show that it appears as the Hessian matrix of the large spin Gaussian approximation of coherent intertwiners, which is the standard method for extracting the semi-classical regime of spinfoam transition amplitudes. This offers an improvement on the usual loop quantum gravity techniques, which mostly focus on the volume operator, in the perspective of modeling (quantum) gravitational waves as shape fluctuations waves propagating on spin network states

Most general theory of 3d gravity: Covariant phase space, dual diffeomorphisms, and more

International audienceWe show that the phase space of three-dimensional gravity contains two layers of dualities: between diffeomorphisms and a notion of “dual diffeomorphisms” on the one hand, and between first order curvature and torsion on the other hand. This is most elegantly revealed and understood when studying the most general Lorentz-invariant first order theory in connection and triad variables, described by the so-called Mielke-Baekler Lagrangian. By analyzing the quasi-local symmetries of this theory in the covariant phase space formalism, we show that in each sector of the torsion/curvature duality there exists a well-defined notion of dual diffeomorphism, which furthermore follows uniquely from the Sugawara construction. Together with the usual diffeomorphisms, these duals form at finite distance, without any boundary conditions, and for any sign of the cosmological constant, a centreless double Virasoro algebra which in the flat case reduces to the BMS$_{3}$ algebra. These algebras can then be centrally-extended via the twisted Sugawara construction. This shows that the celebrated results about asymptotic symmetry algebras are actually generic features of three-dimensional gravity at any finite distance. They are however only revealed when working in first order connection and triad variables, and a priori inaccessible from Chern-Simons theory. As a bonus, we study the second order equations of motion of the Mielke-Baekler model, as well as the on-shell Lagrangian. This reveals the duality between Riemannian metric and teleparallel gravity, and a new candidate theory for three-dimensional massive gravity which we call teleparallel topologically massive gravity

3d gravity in Bondi-Weyl gauge: charges, corners, and integrability

International audienceWe introduce a new gauge and solution space for three-dimensional gravity. As its name Bondi-Weyl suggests, it leads to non-trivial Weyl charges, and uses Bondi-like coordinates to allow for an arbitrary cosmological constant and therefore spacetimes which are asymptotically locally (A)dS or flat. We explain how integrability requires a choice of integrable slicing and also the introduction of a corner term. After discussing the holographic renormalization of the action and of the symplectic potential, we show that the charges are finite, symplectic and integrable, yet not conserved. We find four towers of charges forming an algebroid given by $\mathfrak{vir}\oplus \mathfrak{vir}\oplus$ Heisenberg with three central extensions, where the base space is parametrized by the retarded time. These four charges generate diffeomorphisms of the boundary cylinder, Weyl rescalings of the boundary metric, and radial translations. We perform this study both in metric and triad variables, and use the triad to explain the covariant origin of the corner terms needed for renormalization and integrability