Observables in the Turaev-Viro and Crane-Yetter models

Details Observables in the Turaev-Viro and Crane-Yetter models Observables in the Turaev-Viro and Crane-Yetter models

2004-11-12

arxiv.org/abs/math/0411281v2

J.Math.Phys.48:093508,2007

Mathematics

Quantum Algebra

Approx 26 pages. v2: 4d results substantially extended

Scientific paper

10.1063/1.2759440

We define an invariant of graphs embedded in a three-manifold and a partition function for 2-complexes embedded in a triangulated four-manifold by specifying the values of variables in the Turaev-Viro and Crane-Yetter state sum models. In the case of the three-dimensional invariant, we prove a duality formula relating its Fourier transform to another invariant defined via the coloured Jones polynomial. In the case of the four-dimensional partition function, we give a formula for it in terms of a regular neighbourhood of the 2-complex and the signature of its complement. Some examples are computed which show that the partition function determines an invariant which can detect non locally-flat surfaces in a four-manifold.

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