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Spin foam diagrammatics and topological invariance

Florian Girelli1,2, Robert Oeckl1 and Alejandro Perez1,3

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We provide a simple proof of the topological invariance of the Turaev–Viro model (corresponding to simplicial 3D pure Euclidean gravity with cosmological constant) by means of a novel diagrammatic formulation of the state sum models for quantum BF theories. Moreover, we prove the invariance under more general conditions allowing the state sum to be defined on arbitrary cellular decompositions of the underlying manifold. Invariance is governed by a set of identities corresponding to local gluing and rearrangement of cells in the complex. Due to the fully algebraic nature of these identities our results extend to a vast class of quantum groups. The techniques introduced here could be relevant for investigating the scaling properties of non-topological state sums, proposed as models of quantum gravity in 4D, under refinement of the cellular decomposition.


PACS

04.60.-m Quantum gravity

02.40.Sf Manifolds and cell complexes

02.20.Uw Quantum groups

MSC

57R56 Topological quantum field theories

83Cxx General relativity

17B37 Quantum groups (quantized enveloping algebras) and related deformations (See also 16W35, 20G42, 81R50, 82B23)

57Nxx Topological manifolds

58J70 Invariance and symmetry properties (See also 35A30)

20G42 Quantum groups (quantized function algebras) and their representations (See also 16W35, 17B37, 81R50)

Subjects

Mathematical physics

Gravitation and cosmology

Dates

Issue 6 (21 March 2002)

Received 26 November 2001, in final form 10 January 2002

Published 25 February 2002



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