The Hilbert space of 3d gravity: quantum group symmetries and observables

Meusburger, Catherine; Noui, Karim

doi:10.4310/atmp.2010.v14.n6.a3

Cited by 42 publications

(67 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…On the other hand, the 3d examples here presented are motivated by their role in 3d gravity. In this setting, the quantum group symmetries of quantum homogeneous spaces arise from PoissonLie symmetries in the classical theory and the relevant Poisson-Lie symmetries have been identified in [58,70] as classical doubles. While the case of vanishing cosmological constant is structurally simpler and well understood, 3d anti de Sitter and de Sitter space are of special interest, since they allow one to investigate both, the cosmological constant and Planck's constant as deformation parameters and allow one to understand the role of curvature.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

AdS Poisson homogeneous spaces and Drinfel’d doubles

Ballesteros¹,

Meusburger²,

Naranjo³

2017

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

The correspondence between Poisson homogeneous spaces over a Poisson-Lie group G and Lagrangian Lie subalgebras of the classical double D(g) is revisited and explored in detail for the case in which g = D(a) is a classical double itself. We apply these results to give an explicit description of some coisotropic 2d Poisson homogeneous spaces over the group SL(2, R) ∼ = SO(2, 1), namely 2d anti de Sitter space, 2d hyperbolic space and the lightcone in 3d Minkowski space. We show how each of these spaces is obtained as a quotient with respect to a Poisson-subgroup for one of the three inequivalent Lie bialgebra structures on sl(2, R) and as a coisotropic one for the others.We then construct families of coisotropic Poisson homogeneous structures for 3d anti de Sitter space AdS 3 and show that the ones that are quotients by a Poisson subgroup are determined by a three-parameter family of classical r-matrices for so(2, 2), while the non Poisson-subgroup cases are much more numerous. In particular, we present the two Poisson homogeneous structures on AdS 3 that arise from two Drinfel'd double structures on SO(2, 2). The first one realises AdS 3 as a quotient of SO(2, 2) by the Poisson-subgroup SL(2, R), while the second one, the non-commutative spacetime of the twisted κ-AdS deformation, realises AdS 3 as a coisotropic Poisson homogeneous space.

show abstract

Section: Discussionmentioning

confidence: 99%

“…As shown in [58,70] the relevant Poisson-Lie structures for 3d gravity are classical doubles. Hence, we will restrict attention to quasitriangular and, in particular, classical double Poisson-Lie structures in the following.…”

Section: Ads 3 As a Coisotropic Poisson Homogeneous Space Over A Doublementioning

confidence: 99%

AdS Poisson homogeneous spaces and Drinfel’d doubles

Ballesteros¹,

Meusburger²,

Naranjo³

2017

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The use of the fusion basis emphasizes the Drinfel'd algebra or quantum double structure of (2 + 1) gravity coupled to point defects. This facilitates the comparison with other quantization schemes [93], such as the combinatorial quantization for Chern-Simons theory [94][95][96][97]. Let us also point out the recent work [98,99], which reformulates Kitaev models as a special case of combinatorial quantization via a reformulation of the latter in terms of a Hopf-algebra gauge theory.…”

Section: Jhep02(2017)061mentioning

confidence: 99%

Fusion basis for lattice gauge theory and loop quantum gravity

Delcamp

Dittrich

Riello

2017

J. High Energ. Phys.

133

View full text Add to dashboard Cite

We introduce a new basis for the gauge-invariant Hilbert space of lattice gauge theory and loop quantum gravity in (2 + 1) dimensions, the fusion basis. In doing so, we shift the focus from the original lattice (or spin-network) structure directly to that of the magnetic (curvature) and electric (torsion) excitations themselves. These excitations are classified by the irreducible representations of the Drinfel'd double of the gauge group, and can be readily "fused" together by studying the tensor product of such representations. We will also describe in detail the ribbon operators that create and measure these excitations and make the quasi-local structure of the observable algebra explicit. Since the fusion basis allows for both magnetic and electric excitations from the onset, it turns out to be a precious tool for studying the large scale structure and coarse-graining flow of lattice gauge theories and loop quantum gravity. This is in neat contrast with the widely used spin-network basis, in which it is much more complicated to account for electric excitations, i.e. for Gauß constraint violations, emerging at larger scales. Moreover, since the fusion basis comes equipped with a hierarchical structure, it readily provides the language to design states with sophisticated multi-scale structures. Another way to employ this hierarchical structure is to encode a notion of subsystems for lattice gauge theories and (2 + 1) gravity coupled to point particles. In a follow-up work, we have exploited this notion to provide a new definition of entanglement entropy for these theories.

show abstract

“…Still when Λ = 0, explicit links between LQG and the spinfoam framework [16] or between the Chern Simons combinatorial quantization and LQG [14] have been identified. Note also that we can identify a hidden quantum group structure (the Drinfeld double) in LQG when Λ = 0 [12][13][14], which is consistent with the other approaches. The different cases for 3d gravity are summarized in the first table.…”

Section: Introductionmentioning

confidence: 99%

“…[11] ↔ Turaev-Viro ? ↔ LQG Chern-Simons [12] ↔ Ponzano-Regge [16] ↔ LQG [14] ↔ Chern-Simons Chern-Simons ↔ LQG Chern-Simons [12] ↔ Ponzano-Regge [16] ↔ LQG [14] ↔ Chern-Simons Chern-Simons ? ↔ Turaev-Viro ?…”

Section: Introductionunclassified

Observables in loop quantum gravity with a cosmological constant

Dupuis¹,

Girelli

2014

Phys. Rev. D

View full text Add to dashboard Cite

An open issue in loop quantum gravity (LQG) is the introduction of a non-vanishing cosmological constant Λ. In 3d, Chern-Simons theory provides some guiding lines: Λ appears in the quantum deformation of the gauge group. The Turaev-Viro model, which is an example of spin foam model is also defined in terms of a quantum group. By extension, it is believed that in 4d, a quantum group structure could encode the presence of Λ = 0. In this article, we introduce by hand the quantum group Uq(su(2)) into the LQG framework, that is we deal with Uq(su(2))-spin networks. We explore some of the consequences, focusing in particular on the structure of the observables. Our fundamental tools are tensor operators for Uq(su(2)). We review their properties and give an explicit realization of the spinorial and vectorial ones. We construct the generalization of the U(N ) formalism in this deformed case, which is given by the quantum group Uq(u(n)). We are then able to build geometrical observables, such as the length, area or angle operators ... We show that these operators characterize a quantum discrete hyperbolic geometry in the 3d LQG case. Our results confirm that the use of quantum group in LQG can be a tool to introduce a non-zero cosmological constant into the theory. ContentsIntroduction 2 I. U q (su(2)) in a nutshell 4 A. Definition of U q (su(2)) 4 B. q-harmonic oscillators and the Schwinger-Jordan trick 7II. Tensor operators for U q (su(2)) 7 A. Definition and Wigner-Eckart theorem 7 B. Product of tensor operators: scalar product, vector product and triple product 8 1. Scalar product 9 2. Vector product 10 C. Tensor products of tensor operators 10 III. Realization of tensor operators of rank 1/2 and 1 for U q (su(2)) 11 A. Rank 1/2 tensor operators 11 B. Rank 1 tensor operators 13IV. Observables for the intertwiner space 16 A. General construction and properties of intertwiner observables 16 B. U q (u(n)) formalism for LQG defined over U q (su (2)) 18 [6,7]. Indeed, in a 3d space-time, one can rewrite General Relativity with a (possibly zero) cosmological constant as a Chern-Simons gauge theory 1 . The general phase space structure of the theory for any metric signature and sign of Λ can be treated in a nice unified way [8], using Poisson-Lie groups [9], the classical counterparts of quantum groups. The quantization procedure leads explicitly to a quantum group structure. The full construction, from phase space to quantum group is usually called combinatorial quantization [5][6][7]. We can also quantize 3d gravity using the spinfoam approach. In this approach, 3d gravity is formulated as a BF theory. When Λ = 0, this is the well-known Ponzano-Regge model (both Euclidian or Lorentzian), based on the irreducible unitary representations of the relevant gauge group. When Λ = 0, the quantum group structure is introduced by hand. The Ponzano-Regge model is deformed, using irreducible unitary representations of the relevant quantum deformation of the gauge group. This is then called the Turaev-Viro model [10]. The argument co...

show abstract

The Hilbert space of 3d gravity: quantum group symmetries and observables

Cited by 42 publications

References 49 publications

AdS Poisson homogeneous spaces and Drinfel’d doubles

AdS Poisson homogeneous spaces and Drinfel’d doubles

Fusion basis for lattice gauge theory and loop quantum gravity

Observables in loop quantum gravity with a cosmological constant

Contact Info

Product

Resources

About