Polar duality and the generalized Law of Sines

Kokkendorff, Simon L.

doi:10.1007/s00022-006-1858-7

Cited by 16 publications

(18 citation statements)

References 4 publications

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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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“…The law of sine (49) can be derived by the following consideration [30]. Let d(i) be the distance from the vertex i to the subsimplex (i) and d(i|j) be the distance in the subsimplex (j) of the vertex i to the subsimplex (ij).…”

Section: A Explicit Evaluation Of Qmentioning

confidence: 99%

Linearized dynamics from the 4-simplex Regge action

2007

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We study the relation between the hessian matrix of the riemannian Regge action on a 4-simplex and linearized quantum gravity. We give an explicit formula for the hessian as a function of the geometry, and show that it has a single zero mode. We then use a 3d lattice model to show that (i) the zero mode is a remnant of the continuum diffeomorphism invariance, and (ii) we recover the complete free graviton propagator in the continuum limit. The results help clarify the structure of the boundary state needed in the recent calculations of the graviton propagator in loop quantum gravity, and in particular its role in fixing the gauge.

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“…In the following two subsections we will consider a curved 4-simplex with edge lengths ǫ l ij from which we will derive several quantities for the flat simplex. To do so we will heavily rely on the duality relations of length and angle Gram matrices in the curved case, which are nicely presented in [37]. The desired quantities will be obtained in the limit ǫ → 0.…”

Section: B Limits From Curved Simplexmentioning

confidence: 99%

The Barrett–Crane model: asymptotic measure factor

Kamiński¹,

Steinhaus²

2014

Class. Quantum Grav.

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The original spin foam model construction for 4D gravity by Barrett and Crane suffers from a few troubling issues. In the simple examples of the vertex amplitude they can be summarized as the existence of contributions to the asymptotics from non geometric configurations. Even restricted to geometric contributions the amplitude is not completely worked out. While the phase is known to be the Regge action, the so called measure factor has remained mysterious for a decade. In the toy model case of the 6j symbol this measure factor has a nice geometric interpretation of V −1/2 leading to speculations that a similar interpretation should be possible also in the 4D case. In this paper we provide the first geometric interpretation of the geometric part of the asymptotic for the spin foam consisting of two glued 4-simplices (decomposition of the 4-sphere) in the Barrett-Crane model in the large internal spin regime.

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Polar duality and the generalized Law of Sines

Cited by 16 publications

References 4 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

Linearized dynamics from the 4-simplex Regge action

The Barrett–Crane model: asymptotic measure factor

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