Generating functions for coherent intertwiners

Bonzom, Valentin; Livine, Etera R.

doi:10.1088/0264-9381/30/5/055018

Cited by 32 publications

(66 citation statements)

References 81 publications

(412 reference statements)

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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: 81%

“…To construct coherent boundary spin network states, we use the spinor or holomorphic formalism. This formalism was developed for loop gravity [57][58][59], and led to the definition of coherent intertwiners which can be glued into coherent spin networks [19,43,[60][61][62][63][64]. We denote by |w ∈ C 2 a spinor, by w| its conjugate and by |w] its dual…”

Section: A From Coherent Intertwiners To Coherent Spin Superpositionsmentioning

confidence: 99%

“…Up to the node factorial contribution (J n + 1)!, this weight defines a Poisson distribution on the spins. 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.…”

Section: A From Coherent Intertwiners To Coherent Spin Superpositionsmentioning

confidence: 99%

“…The norm of a coherent intertwiner was computed in [19,62] from its generating function, here keeping the node index n implicit:…”

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: 98%

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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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Section: B Partition Function With Boundary Statesmentioning

confidence: 81%

Section: A From Coherent Intertwiners To Coherent Spin Superpositionsmentioning

confidence: 99%

Section: A From Coherent Intertwiners To Coherent Spin Superpositionsmentioning

confidence: 99%

“…The norm of a coherent intertwiner was computed in [19,62] from its generating function, here keeping the node index n implicit:…”

Section: A From Coherent Intertwiners To Coherent Spin Superpositionsmentioning

confidence: 99%

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confidence: 98%

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

2020

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Duality Between Spin Networks and the 2D Ising Model

Bonzom

Costantino

Livine

2016

Commun. Math. Phys.

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This paper is devoted to studying the relationship between two fundamental objects in physics, associated to finite graphs: the two-dimensional Ising model and the spin network evaluations on planar graphs.Given a graph Γ and a coloring of the edges of Γ, i.e. a map c : E → N where E is the set of edges, the classical spin network evaluation Γ, c is a rational number which is the result of contracting some tensors over irreducible representations of SU(2). Spin networks arise in many areas, in particular related to physics. Since they come from the representation theory of SU(2), they are objects of prime interest in the theory of quantum angular momentum [1] where they are often called Wigner symbols. As such, they have applications in atomic/molecular physics, chemistry, quantum information and so on [2]. More recent applications stem from quantum gravity as spin network equipped with holonomies are the states of loop quantum gravity [3,4] while their evaluations provide quantum gravity amplitudes, known as spin foams, [5][6][7]. The latter are intimately related to lattice topological invariants, such as the Reidemeister torsion [8-10] and the Turaev-Viro invariant of 3-manifolds.Spin network evaluations can be computed in many different ways. The most famous is certainly the combinatorial definition due to R. Penrose [11]. In quantum gravity one often uses contractions of SU(2) intertwiners (notice that it requires an orientation on the edges while Penrose's definition does not -we will prove the equivalence between those evaluations in the main text). In the last years it has been understood [12-16] that a good way to study spin network evaluations on a fixed graph Γ is to organize it in a single generating series Z spin (Γ) := c Γ, c Y c where the symbols Y c stand for a suitable multivariate monomial in formal variables Y e , e ∈ E (full details will be provided later).On a seemingly different side of physics (and mathematical physics), the 2D Ising model can be defined on the same graph Γ. It probably is the most famous statistical model, based on the configuration space of maps from the set V of vertices of Γ into {±1}. The Hamiltonian (energy function) of the model is a sum of interactions between nearest-neighboring sites of Γ, hence associated to the set E of edges of Γ, and weighted by couplings y e , e ∈ E. The partition function Z Ising (Γ) was proved by van den Waerden to be proportional to a sum over even subgraphs of Γ weighted by some monomials in tanh(y e ), e ∈ E(Γ) (see [17] for instance).The present paper is motivated by the following observation made in [21]; if Γ is a planar trivalent graph and for each edge e ∈ E we set Y e := tanh(y e ) then the following equality holds: Unless explicitly stated the contrary we will assume that Γ is planar, i.e. embedded (up to isotopy) in S 2 . This automatically equips Γ with the datum of a cyclic, counter-clockwise ordering of the edges around each vertex. For each angle α around a vertex, following the cyclic ordering around that vertex, we call s(α...

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