Quantum geometry and observables:

Clement Delcamp, Perimeter Institute: New bases for Loop Quantum Gravity

Hilbert spaces of diffeomorphism invariant states can be explicitly realized from TQFTs with defect excitations. Choosing the BF vacuum as the underlying TQFT, we introduce for 2+1D a new basis for kinematical states, namely the fusion basis. This corresponds to a shift of focus from the original lattice to the curvature and torsion excitations themselves. This basis stable under coarse-graining is very well-suited for the study of the large scale limit of the theory and provides a completely relational way of defining regions for 2+1 gravity coupled to particles. Using a Heegaard splitting in order to represent the 3D manifold via a 2D surface, we can lift the 2+1 construction to the 3+1 case. It leads to several bases. In particular, we obtain a basis which naturally encodes curvature degrees of freedom.

Patryk Drobiński, Faculty of Physics, University of Warsaw: The continuum BF vacuum representation

I will present a new representation of the holonomy-flux algebra, inspired by the Pullin-Dittrich-Geiller "BF vacuum" and equivalent to its continuum limit. The work is currently complete for an Abelian (U(1)) gauge group, the SU(2) case still being investigated.

Kristina Giesel, FAU Erlangen-Nürnberg: Cosmological Perturbation Theory with Geometrical Clocks

In this talk we apply the relational formalism to cosmological perturbation theory  and show how geometrical clocks can be obtained that lead to the usual gauge invariant quantities used in the context of linear cosmological perturbations theory such as the Bardeen potentials and the Mukhanov Sasaki variable. We will also compare the framework of reduced phase space quantization using scalar field clocks and geometrical clocks respectively. Finally, we will discuss possible applications of this framework to higher order cosmological perturbation theory.

Florian Girelli, University of Waterloo: A fresh look at 3d quantum gravity

I will focus on 3d gravity (with Λ = 0) and notice that in this case, when discretizing first there exist different interesting choices of polarization: the standard loop gravity choice but also two new ones, the “dual” loop gravity case and the Chern-Simons case. The discretization scheme shows how each choice is related and we also have a clear understanding of how the classical version of a quantum group structure (the Drinfeld double) appears.
I will discuss the quantization of the “dual” loop gravity case and show how it is related to the Dijkgraaf-Witten model. If time allows, I will quickly comment on the 4d extension to this approach and on the common grounds with Dittrich and Geiller’s approach.

Christophe Goeller, Ecole Normale Supérieure de Lyon: Ponzano Regge amplitude on the twisted torus: continuum limit and BMS character formula

We compute the Ponzano-Regge amplitude for 3D quantum gravity on the twisted torus. We compare our result to the previous calcilations done at one-loop for 3D gravity in the asymptotitic limit and in linearized Regge calculus. And we discuss the interpretation of the amplitude as a BMS character

Lucas Hackl, Pennsylvania State University, Institute for Gravitation and the Cosmos: Squeezed vacua in loop quantum

Semi-classical states in quantum gravity are expected to exhibit long range correlations. In order to describe such states within the framework of loop quantum gravity, it is important to parametrize states in terms of their correlations. Recently, we introduced a new class of states with presecribed correlations, called squeezed vacua (see Dr. Yokomizo's talk). In this talk, I will explain node-wise and link-wise squeezing, how the resulting states relate to coherent states studied in the past, and how to encode long-range correlations.

Franz Hinterleitner, Brno University: Canonical LQG operators and kinematical states for plane gravitational waves

In a 1+1 dimensional model of plane gravitational waves the flux-holonomy algebra of loop quantum gravity is modified in such a way that the new basic operators satisfy canonical commutation relations. Thanks to this construction it is possible to find kinematical solutions for unidirectional plane gravitational waves, leading to finite geometric expectation values and fluctuations, which was problematic in an earlier, more conventional approach.

Zichang Huang, Florida Atlantic University: SU(2) Flat Connection on Riemann Surface and Twisted Geometry with a Cosmological Constant

SU(2) flat connection on 2D Riemann surface is shown to relate to the generalized twisted geometry in 3D space with cosmological constant. Various flat connection quantities on Riemann surface are mapped to the geometrical quantities in discrete 3D space. A proposal is that the moduli space of SU(2) flat connections on Riemann surface generalizes the phase space of twisted geometry or Loop Quantum Gravity to include the cosmological constant. I will introduce details on how to construct this map.

Alexander Kegeles, Max Planck Institute for Gravitational Physics: Inequivalent representations of Group Field Theory

Group field theory is a field theoretical formulation of spin networks and simplicial geometry, in which the states are associated with excitations of basic geometrical degrees of freedom over a vacuum of the theory. The simplest type of vacuum is the state of "No geometry". However, that is not the only possible vacuum and there are many more.
In my talk I will present an algebraic formulation of Group Field Theory in which the study of different vacua of the theory can be rigorously addressed and studied. I then show the existence and explicit examples of different inequivalent vacua, based on coherent states of group field theory.

Chun-Yen Lin, University of Warsaw: Quantum reference frames: application in FRW LQC

I will present our recent application of “quantum Cauchy surfaces” to the timeless Dirac theory underlying the standard FRW LQC with a massless scalar field. Two specific foliations, referring to either the scalar or the gravitation sector as the time, generate two complete sets of relational Dirac observables in the timeless physical Hilbert space. These observables then yield two interesting Schr"odinger representations to the Dirac theory. The first representation gives the familiar LQC model describing the quantum gravitational dynamics; the second yields a scalar field quantum dynamics in the background of the discretized spacetime.

Ilkka Mäkinen, University of Warsaw: Spin coherent state representation for intertwiners in loop quantum gravity

We introduce a representation for the intertwiner space in loop quantum gravity, based on projecting intertwiners on coherent states of angular momentum. In this representation, operators such as the Hamiltonian are reformulated as differential operators acting on polynomials of the complex variables which label the unit vectors of the spin coherent states under the stereographic projection. This opens up the possibility of investigating the action of the Hamiltonian geometrically in terms of these unit vectors, as an alternative to calculations relying on standard SU(2) recoupling theory.

Almut Oelmann, University of Erlangen-Nürnberg: Reduced Loop Quantization with four Klein-Gordon Scalar Fields as Reference Matter

We perform a reduced phase space quantization of gravity using four Klein-Gordon scalar fields as reference matter as an alternative to the Brown-Kuchař dust model where eight (dust) scalar fields are used. The scalar fields play the role of so called reference fields that allow to construct Dirac observables for general relativity and introduce a notion of physical spatial and time coordinates. We also compare our results to an earlier model by Domagala et. al. where only one Klein-Gordon scalar field was considered as reference matter for the Hamiltonian constraint. As a result we find that the choice of four Klein-Gordon scalar fields as reference matter leads to a reduced dynamical model that cannot be quantized using loop quantum gravity techniques. However, we further discuss a slight generalization of the action for the four Klein-Gordon scalar fields and show that this leads to a model which can be quantized in the framework of loop quantum gravity.

Julian Rennert, University of Waterloo: Quantization of quasi-Poisson spaces and the Turaev-Viro model

Quantum systems with quasitriangular Hopf algebra symmetry, such as the Turaev-Viro model with real deformation parameter, can be considered as quantizations of phase spaces with Poisson Lie group symmetry. The model 'phase space' for 3D gravity with a positive cosmological constant is the double SU(2)xSU(2), which is not a proper phase space but a quasi-Poisson space and does not have a classical r-matrix. The Turaev-Viro model, however, is defined in terms of certain representations of a Hopf algebra at q root of unity. Following work by G. Mack and V. Schomerus we will argue that it is more consistent with our classical setting to consider the Turaev-Viro model in terms of a quasi-Hopf algebra. Within this setting we can construct certain geometric operators probing the quantum geometry and a Hamiltonian constraint whose kernel gives rise to the Turaev-Viro amplitudes.

Johannes Thürigen, Laboratoire de Physique Théorique, Université Paris-Sud XI: The spectral dimension of spin-foam spacetime

The spectral dimension has proven to be a very informative observable to understand the properties of quantum geometries in approaches to quantum gravity. In Spin-foam quantum gravity, explicit evaluation of physically relevant observables remains one of the most pressing issues.
As a first step towards determining the spectral dimension of spin-foam quantum spacetime, we provide the framework for its definition and various methods for calculation, based both on analytical results as well as numerical (Monte-Carlo) methods. Furthermore, we perform the computation of the spectral dimension of spacetime in the simplified cuboid spin-foam model.
We find evidence for a generic flow of the spacetime dimension which can be related to the renormalization group flow as obtained in spin-foam coarse graining.

Marko Vojinovic, Institute of Physics, University of Belgrade, Serbia: Categorical generalization of spinfoam models --- review and applications

We will give a review of the construction and applications of a spinfoam-like model based on a 2-group, a categorical generalization of the notion of a group. Starting from the Poincare 2-group, one can construct the so-called spincube model of quantum gravity, which features labeled edges of a triangulation, thus allowing for straightforward coupling of matter fields. The coupling of matter fields to the spincube model enables us to apply it to various interesting topics, such as the study of the cosmological constant, the symmetry-protected entanglement between gravity and matter, and grand-unification schemes. We will discuss each of these applications in turn, and give prospects for future research.

Mingyi Zhang, Albert Einstein Institute: Holographic entropy from group field theory

I will present recent results on calculating the holographic entanglement entropy from the group field theory. I will show a dictionary between group field theory (thus, spin networks and random tensors) states and generalized random tensor networks. Then, I will present how to use this dictionary to compute the R'{e}nyi entropy of such states and recover the Ryu-Takayanagi formula, in three different cases corresponding to three different truncations/approximations, suggested by the established correspondence.