Invariants of Spin Networks Embedded in Three-Manifolds

Details Invariants of Spin Networks Embedded in Three-Manifolds Invariants of Spin Networks Embedded in Three-Manifolds

2006-12-21

arxiv.org/abs/gr-qc/0612137v2

Commun.Math.Phys.279:381-399,2008

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

29 pages, 9 figures; v2 small changes in the abstract

Scientific paper

10.1007/s00220-008-0422-8

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.

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