Closure constraints for hyperbolic tetrahedra

Charles, Christoph; Livine, Etera R.

doi:10.1088/0264-9381/32/13/135003

Cited by 14 publications

(39 citation statements)

References 41 publications

(102 reference statements)

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“…A possible reason could be polar duality [76]. Another possible link is the interpretation of the Gauß constraints (or closure constraints) as a Bianchi identity and a possible re-construction of a new kind of connection proposed in [86,87].…”

Section: Jhep05(2017)123mentioning

confidence: 99%

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

Dittrich

2017

J. High Energ. Phys.

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We apply the recently suggested strategy to lift state spaces and operators for (2 + 1)-dimensional topological quantum field theories to state spaces and operators for a (3 + 1)-dimensional TQFT with defects. We start from the (2 + 1)-dimensional TuraevViro theory and obtain a state space, consistent with the state space expected from the Crane-Yetter model with line defects.This work has important applications for quantum gravity as well as the theory of topological phases in (3 + 1) dimensions. It provides a self-dual quantum geometry realization based on a vacuum state peaked on a homogeneously curved geometry. The state spaces and operators we construct here provide also an improved version of the Walker-Wang model, and simplify its analysis considerably.We in particular show that the fusion bases of the (2 + 1)-dimensional theory lead to a rich set of bases for the (3 + 1)-dimensional theory. This includes a quantum deformed spin network basis, which in a loop quantum gravity context diagonalizes spatial geometry operators. We also obtain a dual curvature basis, that diagonalizes the Walker-Wang Hamiltonian.Furthermore, the construction presented here can be generalized to provide state spaces for the recently introduced dichromatic four-dimensional manifold invariants.

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Section: Jhep05(2017)123mentioning

confidence: 99%

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

Dittrich

2017

J. High Energ. Phys.

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“…Hence, we effectively obtain torsion, defined as a violation of the Gauß constraint, due to the presence of curvature. Such an effect, which was named curvature-induced torsion in [14], is strictly related to the need of deforming the Gauß constraint in phase spaces describing piecewise homogeneouslycurved (instead of piecewise flat) geometries [55,[79][80][81][82][83] (see also [84][85][86][87], for an analysis in four dimensions). In terms of defect excitations discussed in this paper, torsion excitations interpreted as spinning particles can arise from the fusion of two spinless defects, since two particles can have orbital angular momentum.…”

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

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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.

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“…As pointed out in [28,32,33], these boosts satisfy an almost-closure relation around the triangle, in that their product is the identity up to now a possible rotation:…”

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confidence: 93%

“…This sets our goal: writing down SB(2, C) elements that depends only on the geometrical data of faces of homogeneously curved polyhedron such that it satisfies the (generalized) closure condition and it has a natural interpretation as a normal. We will construct such structures in this paper.We actually offered such a proposal in [32]. But in the construction, we only used the closure constraint as a guiding line.…”

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confidence: 99%

“…Moreover, we get a whole one-parameter family of closure constraints. This means that one has to first decide the value of the parameter β before reconstructing the hyperbolic tetrahedron from the non-abelian normals L i .As for the SU(2) closure constraint, it proposes a generalization of the SU(2) closure constraints for hyperbolic tetrahedra derived in [32,33]. Indeed these recent studies relied on using the holonomies of the spin-connection to define SU(2) group elements normal to each hyperbolic triangle of the tetrahedron.…”

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confidence: 99%

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The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

Charles

Livine

2017

Gen Relativ Gravit

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The closure constraint is a central piece of the mathematics of loop quantum gravity. It encodes the gauge invariance of the spin network states of quantum geometry and provides them with a geometrical interpretation: each decorated vertex of a spin network is dual to a quantized polyhedron in R 3 . For instance, a 4-valent vertex is interpreted as a tetrahedron determined by the four normal vectors of its faces. We develop a framework where the closure constraint is re-interpreted as a Bianchi identity, with the normals defined as holonomies around the polyhedron faces of a connection (constructed from the spinning geometry interpretation of twisted geometries). This allows us to define closure constraints for hyperbolic tetrahedra (living in the 3-hyperboloid of unit future-oriented spacelike vectors in R 3,1 ) in terms of normals living all in SU(2) or in SB(2, C). The latter fits perfectly with the classical phase space developed for q-deformed loop quantum gravity supposed to account for a non-vanishing cosmological constant Λ > 0. This is the first step towards interpreting q-deformed twisted geometries as actual discrete hyperbolic triangulations. IntroductionLoop quantum gravity (for monographs see [1][2][3]) is based on a reformulation of general relativity as an SU (2) gauge theory of the Ashtekar-Barbero connection [4][5][6][7] together with its canonically conjugated field, the densitized triad 1 . At the quantum level, the kinematics of the theory is well-understood: the space of quantum states of geometry are wave-functions of the Ashtekar-Barbero connection and admits a canonical basis labelled by spin networks [8]. Spin networks are (abstract) graphs (which might have knotting information in 3+1d) colored with spins on the edges (hence the name) and intertwiners, which are invariant representations of the gauge group, on the vertices. The spin network basis diagonalize the geometrical operators. The spins carried by the edges corresponding to quanta of surfaces [9]. The volume operator is a bit more involved but we still can find a basis of intertwiners that diagonalize the volume operator.The spin network states have a natural geometrical interpretation in the twisted geometry framework [12]. Each node of the graph is interpreted as a flat convex polyhedron. Each edge coming out of the vertex corresponds to a face of the polyhedron and the area of this face is given by the spin carried by the edge. The intertwiner should encode the remaining quantum degrees of freedom. Then, the polyhedra corresponding to the vertices are glued together on their faces. By construction, the area of two glued faces match. Still, the geometry is said to be twisted since the shapes of the faces do not have to match. In the continuum, this corresponds to a discontinuous metric [13]. There is a notable second geometrical parametrization that was proposed: spinning geometries [14]. The construction is roughly the same but the faces and edges of the polyhedra are not assumed to be (extrinsically) flat. As a consequence,...

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Closure constraints for hyperbolic tetrahedra

Cited by 14 publications

References 41 publications

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

Fusion basis for lattice gauge theory and loop quantum gravity

The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

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