Invariants of Spin Networks Embedded in Three-Manifolds

TL;DR

This paper establishes that spin network invariants derived from SU(2) BF-Theory are topological invariants, unifying various approaches and relating them to Chern-Simons invariants using the Chain-Mail technique.

Contribution

It introduces a general definition of BF-Theory spin network invariants via Chain-Mail, proving their topological nature and connecting them with existing invariants like Witten-Reshetikhin-Turaev and Turaev's invariants.

Findings

BF-Theory invariants are topological invariants.

The invariants coincide with Turaev's invariants for colored graphs.

A unified framework for spin network invariants is provided.

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.