q-deformed LQG in three dimensional space-time

Loop quantum gravity (LQG) is a canonical, background-independent and non-perturbative approach to quantum gravity. This thesis is devoted to studying three-dimensional (3D) quantum gravity with a non-vanishing cosmological constant Λ in the LQG approach. In particular, we focus on the case of Λ negative in the Euclidean signature where the isometry group is SL(2, C). We construct the q-deformed LQG model with the real deformation parameter q encoding the value of | Λ | . In this model, the kinematical and physical Hilbert spaces of gravity exhibit the quantum group symmetries consistent with other approaches to 3D quantum gravity. The LQG model with Λ = 0 is recovered at q → 1. This quantum gravity model is derived from the classical theory using the standard canonical quantization program a` la Dirac and the mathematical connection between quantum groups and Lie bialgebras. We establish this model first in terms of the holonomy-flux algebra and then in terms of spinors, which are purely geometrical objects. We write the quantum Hamiltonian constraint with spinors and recover the Turaev-Viro amplitudes defined in the spinfoam model, which is a covariant approach to quantum gravity written in a discrete path integral formalism. We also use spinors to reconstruct the spinfoam model in a way that the local building blocks to construct global 3D geometry are conformal. This is done for the Λ = 0 case as a first step. The q-deformed LQG model is topological and describes the curved geometries in 3D as well as on 2D spatial surfaces. It is expected to serve as a better starting point to connect LQG with other quantum gravity approaches and generalize to 4D LQG with a non-vanishing Λ

Local Observables in SU⁡q(2)\operatorname{SU}_q(2) Lattice Gauge Theory

We consider a deformation of 3D lattice gauge theory in the canonical picture, first classically, based on the Heisenberg double of $\operatorname{SU}(2)$, then at the quantum level. We show that classical spinors can be used to define a fundamental set of local observables. They are invariant quantities which live on the vertices of the lattice and are labelled by pairs of incident edges. Any function on the classical phase space, e.g. Wilson loops, can be rewritten in terms of these observables. At the quantum level, we show that spinors become spinor operators. The quantization of the local observables then requires the use of the quantum $\mathcal{R}$-matrix which we prove to be equivalent to a specific parallel transport around the vertex. We provide the algebra of the local observables, as a Poisson algebra classically, then as a $q$-deformation of $\mathfrak{so}^*(2n)$ at the quantum level. This formalism can be relevant to any theory relying on lattice gauge theory techniques such as topological models, loop quantum gravity or of course lattice gauge theory itself.Comment: 36+4 pages, 9 figures; updated version submitted to journa

3D Quantum Gravity from Holomorphic Blocks

Three-dimensional gravity is a topological field theory, which can be quantized as the Ponzano-Regge state-sum model built from the $\{3nj\}$-symbols of the recoupling of the \SU(2) representations, in which spins are interpreted as quantized edge lengths in Planck units. It describes the flat spacetime as gluing of three-dimensional cells with a fixed boundary metric encoding length scale. In this paper, we revisit the Ponzano-Regge model formulated in terms of spinors and rewrite the quantum geometry of 3D cells with holomorphic recoupling symbols. These symbols, known as Schwinger's generating function for the $\{6j\}$-symbols, are simply the squared inverse of the partition function of the 2D Ising model living on the boundary of the 3D cells. They can furthermore be interpreted, in their critical regime, as scale-invariant basic elements of geometry. We show how to glue them together into a discrete topological quantum field theory. This reformulation of the path integral for 3D quantum gravity, with a rich pole structure of the elementary building blocks, opens a new door toward the study of phase transitions and continuum limits in 3D quantum gravity, and offers a new twist on the construction of a duality between 3D quantum gravity and a 2d conformal theory

Spinor Representation of the Hamiltonian Constraint in 3D LQG with a Non-zero Cosmological Constant

We develop in a companion article the kinematics of three-dimensional loop quantum gravity in Euclidean signature and with a negative cosmological constant, focusing in particular on the spinorial representation which is well-known at zero cosmological constant. In this article, we put this formalism to the test by quantizing the Hamiltonian constraint on the dual of a triangulation. The Hamiltonian constraints are obtained by projecting the flatness constraints onto spinors, as done in the flat case by the first author and Livine. Quantization then relies on $q$-deformed spinors. The quantum Hamiltonian constraint acts in the $q$-deformed spin network basis as difference equations on physical states, which are thus the Wheeler-DeWitt equations in this framework. Moreover, we study how physical states transform under Pachner moves of the canonical surface. We find that those transformations are in fact $q$-deformations of the transition amplitudes of the flat case as found by Noui and Perez. Our quantum Hamiltonian constraints therefore build a Turaev-Viro model at real $q$

qq-deformed 3D Loop Gravity on the Torus

International audienceThe q-deformed loop gravity framework was introduced as a canonical formalism for the Turaev–Viro model (with ), allowing to quantize 3D Euclidean gravity with a (negative) cosmological constant using a quantum deformation of the gauge group. We describe its application to the 2-torus, explicitly writing the q-deformed gauge symmetries and deriving the reduced physical phase space of Dirac observables, which leads back to the Goldman brackets for the moduli space of flat connections. Furthermore it turns out that the q-deformed loop gravity can be derived through a gauge fixing from the Fock–Rosly bracket, which provides an explicit link between loop quantum gravity (for q real) and the combinatorial quantization of 3D gravity as a Chern–Simons theory with non-vanishing cosmological constant . A side-product is the reformulation of the loop quantum gravity phase space for vanishing cosmological constant , based on holonomies and fluxes, in terms of Poincaré holonomies. Although we focus on the case of the torus as an example, our results outline the general equivalence between 3D q-deformed loop quantum gravity and the combinatorial quantization of Chern–Simons theory for arbitrary graph and topology

Local Observables in SU⁡q(2)\operatorname{SU}_q(2) Lattice Gauge Theory

We consider a deformation of 3D lattice gauge theory in the canonical picture, first classically, based on the Heisenberg double of $\operatorname{SU}(2)$, then at the quantum level. We show that classical spinors can be used to define a fundamental set of local observables. They are invariant quantities which live on the vertices of the lattice and are labelled by pairs of incident edges. Any function on the classical phase space, e.g. Wilson loops, can be rewritten in terms of these observables. At the quantum level, we show that spinors become spinor operators. The quantization of the local observables then requires the use of the quantum $\mathcal{R}$-matrix which we prove to be equivalent to a specific parallel transport around the vertex. We provide the algebra of the local observables, as a Poisson algebra classically, then as a $q$-deformation of $\mathfrak{so}^*(2n)$ at the quantum level. This formalism can be relevant to any theory relying on lattice gauge theory techniques such as topological models, loop quantum gravity or of course lattice gauge theory itself