Observables in the Turaev-Viro and Crane-Yetter models
John W. Barrett, J. Manuel Garcia-Islas, Joao Faria Martins
TL;DR
The work constructs quantum-topological observables in the Turaev–Viro and Crane–Yetter state-sum frameworks by labeling subcomplexes of 3- and 4-manifolds. It establishes a Fourier-duality between the 3D graph observable and the relativistic spin-network invariant, and provides an explicit 4D observable formula expressed through a regular neighbourhood and the complement’s signature, enabling computation via the Witten–Reshetikhin–Turaev invariants of boundary links. It demonstrates nontrivial detection of non locally-flat embeddings and shows how trivial-labelings recover known invariants, while chain-mail and shadow-world techniques underpin the connections to Kirby calculus and 3D/4D topological quantum field theories. The results bridge graph observables in TV/CY models with established quantum invariants, offering computationally accessible expressions and potential applications to quantum gravity models. Collectively, the paper advances the use of chain-mail, Fourier transforms, and WRT-type invariants to define and evaluate observables in low-dimensional quantum topology and quantum gravity contexts.
Abstract
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.