Four-dimensional quantum gravity with a cosmological constant from three-dimensional holomorphic blocks

Haggard, Hal M.; Han, Muxin; Kamiński, Wojciech; Riello, Aldo

doi:10.1016/j.physletb.2015.11.058

Cited by 48 publications

(135 citation statements)

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“…The WTR invariant can be considered as a quantization of three-dimensional ChernSimons theory. This brings us to recent work by Haggard, Han, Kaminski and Riello [40][41][42][43][44], which construct spin foam amplitudes for four-dimensional building blocks via the Chern-Simons theory defined on the boundary of these building blocks. The work [40][41][42][43][44] also provides a semi-classical analysis of these amplitudes leading to a phase space describing blocks of homogeneously curved geometry.…”

Section: Jhep05(2017)123mentioning

confidence: 92%

“…In this work we will provide a quantization of this phase space. We will encounter more similarities with [40][41][42][43][44]: namely longitudinal and transversal Wilson loops (or holonomies in [40][41][42][43][44]) that take over the role of both holonomies and conjugated fluxes. 'Longitudinal' and 'transversal' are with respect to the boundary of a blow-up of the one-skeleton of the dual to the triangulation.…”

Section: Jhep05(2017)123mentioning

confidence: 99%

“…The spin network basis diagonalizes Wilson loops along the boundary of the triangles of the triangulation, which represent exponentiated flux operators [15][16][17][40][41][42][43][44]. The curvature basis diagonalizes Wilson loops around the edges of the triangulation and represent holonomy operators, measuring curvature.…”

Section: Jhep05(2017)123mentioning

confidence: 99%

“…Another reason to identify the Wilson loop around triangles with an exponentiated area operator is the analysis of [40][41][42][43][44]. There one considers phase spaces associated to homogeneously curved simplices.…”

Section: Operators On the Constrained Hilbert Spacementioning

confidence: 99%

“…Such a dual structure of holonomy operators, around both edges and triangles of a 4-simplex, plays also an important role in [40][41][42][43][44], which constructs the 4-simplex amplitude for a spin foam model describing gravity with a cosmological constant, and examines the semi-classical limit of this amplitude. The work [40][41][42][43][44] also discusses the phase space associated to the boundary of a homogeneously curved simplex, and notes the dual role of the two kinds of holonomies.…”

Section: The 4-simplex Triangulation Of the 3-spherementioning

confidence: 99%

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(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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