On the exact evaluation of spin networks

Freidel, Laurent; Hnybida, Jeff

doi:10.1063/1.4830008

Cited by 23 publications

(75 citation statements)

References 49 publications

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“…A useful reparametrization of the twisted geometry phase space is in terms of spinor networks [9-11, 27, 28]. This spinorial parametrization of twisted geometries led to a systematic construction of coherent intertwiners [7,8] and semi-classical spin network states [29,30] and remarkable exact computations of spinfoam transition amplitudes [31,32]. One introduces a complex 2-vector, or spinor, z v e ∈ C 2 on each half-edge.…”

Section: Discrete Bubble Networkmentioning

confidence: 99%

Bubble networks: framed discrete geometry for quantum gravity

Freidel

Livine

2018

Gen Relativ Gravit

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In the context of canonical quantum gravity in 3+1 dimensions, we introduce a new notion of bubble network that represents discrete 3d space geometries. These are natural extensions of twisted geometries, which represent the geometrical data underlying loop quantum geometry and are defined as networks of SU(2) holonomies. In addition to the SU(2) representations encoding the geometrical flux, the bubble network links carry a compatible SL(2, R) representation encoding the discretized frame field which composes the flux. In contrast with twisted geometries, this extra data allows to reconstruct the frame compatible with the flux unambiguously. At the classical level this data represents a network of 3d geometrical cells glued together. The SL(2, R) data contains information about the discretized 2d metrics of the interfaces between 3d cells and SL(2, R) local transformations are understood as the group of area-preserving diffeomorphisms. We further show that the natural gluing condition with respect to this extended group structure ensures that the intrinsic 2d geometry of a boundary surface is the same from the viewpoint of the two cells sharing it. At the quantum level this gluing corresponds to a maximal entanglement along the network edges. We emphasize that the nature of this extension of twisted geometries is compatible with the general analysis of gauge theories that predicts edge mode degrees of freedom at the interface of subsystems.

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Section: Discrete Bubble Networkmentioning

confidence: 99%

Bubble networks: framed discrete geometry for quantum gravity

Freidel

Livine

2018

Gen Relativ Gravit

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“…More specifically, we have in mind the spin foam framework, whose goal is to define a path integral for quantum gravity formulated using tools from topological quantum field theory [24][25][26][27][28][29][30][31]. In fact, for spin foams, the case of a vanishing cosmological constant is particularly well understood and many techniques for the analyses of the spin foam partition functions are available, especially in three dimensions [32][33][34][35][36][37][38][39][40]. In the three-dimensional case, the spin foam model goes back to a clever observation by Ponzano and Regge [41] which led to what can be considered as the earliest proposal for a 3D quantum gravity theory.…”

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confidence: 99%

Quasi-local holographic dualities in non-perturbative 3d quantum gravity II – From coherent quantum boundaries to BMS3 characters

et al. 2019

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We analyze the partition function of three-dimensional quantum gravity on the twisted solid tours and the ensuing dual field theory. The setting is that of a non-perturbative model of three dimensional quantum gravity-the Ponzano-Regge model, that we briefly review in a self-contained manner-which can be used to compute quasi-local amplitudes for its boundary states. In this second paper of the series, we choose a particular class of boundary spin-network states which impose Gibbons-Hawking-York boundary conditions to the partition function. The peculiarity of these states is to encode a two-dimensional quantum geometry peaked around a classical quadrangulation of the finite toroidal boundary. Thanks to the topological properties of three-dimensional gravity, the theory easily projects onto the boundary while crucially still keeping track of the topological properties of the bulk. This produces, at the non-perturbative level, a specific non-linear sigma-model on the boundary, akin to a Wess-Zumino-Novikov-Witten model, whose classical equations of motion can be used to reconstruct different bulk geometries: the expected classical one is accompanied by other "quantum" solutions. The classical regime of the sigma-model becomes reliable in the limit of large boundary spins, which coincides with the semiclassical limit of the boundary geometry. In a 1-loop approximation around the solutions to the classical equations of motion, we recover (with corrections due to the non-classical bulk geometries) results obtained in the past via perturbative quantum General Relativity and through the study of characters of the BMS 3 group. The exposition is meant to be completely 11 This pair of variables is the one identified by Kapovich and Millson [77]. 12 Indeed, in the 3-dimensional context in which they were originally developed, such intertwiners are interpreted as quantum tetrahedra in the four-valent case m = 4, and more generally as quantum polyhedra for higher valencies [80][81][82][83]. See also [78,84] for a discussion of polyhedra in homogeneously curved space. In the context of canonical 3d gravity, the interpretation of SU(2) intertwiners as polygons was never truly developed beyond the interpretation of 3-valent intertwiners as quantum triangles. 13 If non-closing configuration are excluded, their contribution is suppressed in the large-spin limit [32], which admits in turn a semiclassical interpretation in terms of Regge geometries.14 Computing the expectation values of the su(2) generators on the state |j, j , we get J = (0, 0, j). In turn, we can compute the variance J 2 = j(j + 1), which is simply given by the su(2) Casimir. Thus the state |j, j corresponds to a semi-classical vector of length j in the z-direction, peaked on (0, 0, j) with spread 1 J J 2 − J 2 ∼ 1 √ j . The corresponding polar angle θ can be estimated to be θ ≈ arccos j √ j(j+1) ≈ 1 √ j → 0. 15 Indeed, the S2 in which n lives is better understood as SU(2)/U(1). This corresponds precisely to the Hopf fibration S2 ∼ = S3/S1.

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“…It has been further proved that the generating functions for these "planar" Ponzano-Regge amplitudes for a 3-valent boundary graph Γ are dual to the 2D Ising model living on the 2D cellular complex [20,53]. This duality involves the same class of coherent spin network states that we will use in the present work and that have been shown to allow for exact analytic computations of spinfoam amplitudes [18,19,54]. Finally, general results for the Ponzano-Regge partition function for the 3-sphere and handlebodies can be found in [13,14,55].…”

Section: B Partition Function With Boundary Statesmentioning

confidence: 79%

“…Such coherent states have the natural mathematical interpretation as a generating function for LS spin networks [19,43,66]. The sum over the spins {j l } defines a series with a non-zero radius of convergence [18] and the boundary state is defined outside of the radius of convergence as the analytic continuation of the series. Such a coherent state is interpreted as a superposition of discrete geometries on the 2D boundary with different edge lengths [20].…”

Section: A From Coherent Intertwiners To Coherent Spin Superpositionsmentioning

confidence: 99%

“…Following this line of research developed in [1][2][3]17], we propose here to use a class of coherent boundary states, defined as quantum superpositions of spins-thus lengths-admitting a critical regime with a manifest scale invariance in the semi-classical limit. Introduced in [18][19][20][21], we show that these coherent spin network wave-packets allow for an exact evaluation of the Ponzano-Regge amplitudes with quantum boundary. Applying this to the case of a solid twisted torus, relevant to the AdS 3 /CFT 2 correspondence and the BTZ black hole, we recover a regularized version of the character formula for the Bondi-Metzner-Sachs (BMS) group formally defined as a modular form in terms of the Dedekind η-function.…”

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confidence: 97%

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Non-perturbative 3D quantum gravity: quantum boundary states and exact partition function

2020

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We push forward the investigation of holographic dualities in 3d quantum gravity formulated as a topological quantum field theory, studying the correspondence between boundary and bulk structures. Working with the Ponzano-Regge topological state-sum model defining an exact discretization of 3d quantum gravity, we analyze how the partition function for a solid twisted torus depends on the boundary quantum state. This configuration is relevant to the AdS3/CFT2 correspondence. We introduce boundary spin network states with coherent superposition of spins on a square lattice on the boundary surface. This allows for the first exact analytical calculation of Ponzano-Regge amplitudes with extended 2D boundary (beyond the single tetrahedron). We get a regularized finite truncation of the BMS character formula obtained from the one-loop perturbative quantization of 3d gravity. This hints towards the existence of an underlying symmetry and the integrability of the theory for finite boundary at the quantum level for coherent boundary spin network states. Contents Acknowledgement 38A. Boundary linear forms on the 2-torus 38 B. Stationary spin configurations from Critical couplings 40 C. Ponzano-Regge determinants and bi-variate Chebyshev polynomials 41References 42 * Electronic address: christophe.goeller@ens-lyon.fr † Electronic address: etera.livine@ens-lyon.fr ‡ Electronic address: ariello@perimeterinstitute.ca 5 A spin network mathematically refers to the basis of the space of SU(2)-gauge invariant L 2 wave-functions diagonalizing the SU(2) Casimirs and constructed from SU(2) Wigner matrices and intertwiners [49,50].However, it is often used to indicate more broadly a generic wave-function in that Hilbert space. For a discussion of the gauge invariance of spin networks and their extensions under SU(2) transformations, the interested reader can refer to [51].

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On the exact evaluation of spin networks

Cited by 23 publications

References 49 publications

Bubble networks: framed discrete geometry for quantum gravity

Bubble networks: framed discrete geometry for quantum gravity

Quasi-local holographic dualities in non-perturbative 3d quantum gravity II – From coherent quantum boundaries to BMS3 characters

Non-perturbative 3D quantum gravity: quantum boundary states and exact partition function

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