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Communications in Mathematical Physics

, Volume 279, Issue 2, pp 381–399 | Cite as

Invariants of Spin Networks Embedded in Three-Manifolds

  • João Faria Martins
  • Aleksandar Miković
Article

Abstract

We study the invariants of spin networks embedded in a three-dimensional manifold which are based on the path integral for SU(2) BF-Theory. These invariants appear naturally in Loop Quantum Gravity, and have been defined as spin-foam state sums. By using the Chain-Mail technique, we give a more general definition of these invariants, and show that the state-sum definition is a special case. This provides a rigorous proof that the state-sum invariants of spin networks are topological invariants. We derive various results about the BF-Theory spin network invariants, and we find a relation with the corresponding invariants defined from Chern-Simons Theory, i.e. the Witten-Reshetikhin-Turaev invariants. We also prove that the BF-Theory spin network invariants coincide with V. Turaev’s definition of invariants of coloured graphs embedded in 3-manifolds and thick surfaces, constructed by using shadow-world evaluations. Our framework therefore provides a unified view of these invariants.

Keywords

Quantum Gravity Quantum Group Orientation Preserve Loop Quantum Gravity Spin Network 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Departamento de MatemáticaInstituto Superior Técnico (Universidade Técnica de Lisboa)LisboaPortugal
  2. 2.Departamento de MatemáticaUniversidade Lusófona de Humanidades e TecnologiaLisboaPortugal
  3. 3.Grupo de Física Matemática da Universidade de LisboaLisboaPortugal

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