Holography for the Lorentz group Racah coefficients

Krasnov, Kirill

doi:10.1088/0264-9381/22/11/003

Cited by 3 publications

(2 citation statements)

References 10 publications

(16 reference statements)

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“…[79]) might bring further inspiration towards this goal. Second, we would like to take the present work as a first step towards revisiting the link between 3d quantum gravity and 2d CFT [80][81][82][83][84]. For instance, starting from the discrete boundary current symmetry algebra should most certainly help eludicate the arising of (regularized) BMS character from Ponzano-Regge amplitudes as shown in [63][64][65][66] and help understand the conformal invariance and criticality at the finite discrete level for quasi-local boundaries in 3d quantum gravity.…”

Section: Discussionmentioning

confidence: 99%

The Quantum Gravity Disk: Discrete Current Algebra

Freidel,

Goeller,

Livine

2021

Preprint

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We study the quantization of the corner symmetry algebra of 3d gravity associated with 1d spatial boundaries. We first recall that in the continuum, this symmetry algebra is given by the central extension of the Poincaré loop algebra. At the quantum level, we construct a discrete current algebra with a DSU(2) quantum symmetry group that depends on an integer N . This algebra satisfies two fundamental properties: First it is compatible with the quantum space-time picture given by the Ponzano-Regge state-sum model, which provides a path integral amplitudes for 3d loop quantum gravity. We then show that we recover in the N → ∞ limit the central extension of the Poincaré current algebra. The number of boundary edges defines a discreteness parameter N which counts the number of flux lines attached to the boundary. We analyse the refinement, coarse-graining and fusion processes as N changes. Identifying a discrete current algebra on quantum boundaries is an important step towards understanding how conformal field theories arise on spatial boundaries in loop quantum gravity. It also shows how an asymptotic BMS symmetry group could appear from the continuum limit of 3d quantum gravity. Contents B. Commutators of the tentative discrete Virasoro operators Dn

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Section: Discussionmentioning

confidence: 99%

The Quantum Gravity Disk: Discrete Current Algebra

Freidel,

Goeller,

Livine

2021

Preprint

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“…To obtain an amplitude for a manifold one multiplies these wavefunctions and integrates over the gluing variables x, y. We would like to note that there is a similarity in the structure of the vertex found and the expression for the 6j -symbol obtained in [21]. In both cases the quantity in question has to do with a truncated tetrahedron, and is constructed using propagators for long edges (the second set of δ-functions in (6.9)), and a set of factors for the small faces (the first set of δ-functions in (6.9)).…”

Section: Group Field Theory Feynman Diagram Analysismentioning

confidence: 96%

Quantum gravity with matter and group field theory

Krasnov¹

2007

Class. Quantum Grav.

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Abstract. A generalization of the matrix model idea to quantum gravity in three and higher dimensions is known as group field theory (GFT). In this paper we study generalized GFT models that can be used to describe 3D quantum gravity coupled to point particles. The generalization considered is that of replacing the group leading to pure quantum gravity by the twisted product of the group with its dual -the so-called Drinfeld double of the group. The Drinfeld double is a quantum group in that it is an algebra that is both noncommutative and non-cocommutative, and special care is needed to define group field theory for it. We show how this is done, and study the resulting GFT models. Of special interest is a new topological model that is the "Ponzano-Regge" model for the Drinfeld double. However, as we show, this model does not describe point particles. Motivated by the GFT considerations, we consider a more general class of models that are defined using not GFT, but the so-called chain mail techniques. A general model of this class does not produce 3-manifold invariants, but has an interpretation in terms of point particle Feynman diagrams.

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The quantum gravity disk: Discrete current algebra

Freidel

Goeller

Livine

2021

Journal of Mathematical Physics

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We study the quantization of the corner symmetry algebra of 3D gravity, that is, the algebra of observables associated with 1D spatial boundaries. In the continuum field theory, at the classical level, this symmetry algebra is given by the central extension of the Poincaré loop algebra. At the quantum level, we construct a discrete current algebra based on a quantum symmetry group given by the Drinfeld double DSU(2). Those discrete currents depend on an integer N, a discreteness parameter, understood as the number of quanta of geometry on the 1D boundary: low N is the deep quantum regime, while large N should lead back to a continuum picture. We show that this algebra satisfies two fundamental properties. First, it is compatible with the quantum space-time picture given by the Ponzano–Regge state-sum model, which provides discrete path integral amplitudes for 3D quantum gravity. The integer N then counts the flux lines attached to the boundary. Second, we analyze the refinement, coarse-graining, and fusion processes as N changes, and we show that the N → ∞ limit is a classical limit where we recover the Poincaré current algebra. Identifying such a discrete current algebra on quantum boundaries is an important step toward understanding how conformal field theories arise on spatial boundaries in quantized space-times such as in loop quantum gravity.

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Holography for the Lorentz group Racah coefficients

Cited by 3 publications

References 10 publications

The Quantum Gravity Disk: Discrete Current Algebra

The Quantum Gravity Disk: Discrete Current Algebra

Quantum gravity with matter and group field theory

The quantum gravity disk: Discrete current algebra

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