Communications in Mathematical Physics

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

Invariants of Spin Networks Embedded in Three-Manifolds

  • João Faria MartinsEmail author
  • Aleksandar Miković


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.


Quantum Gravity Quantum Group Orientation Preserve Loop Quantum Gravity Spin Network 
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© 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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