Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion

Abstract

We consider quantum invariants of 3-manifolds associated with arbitrary simple Lie algebras. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant corresponding to the projective group. We then show that the projective quantum invariant is always an algebraic integer, if the quantum parameter is a prime root of unity. We also show that the projective quantum invariant of rational homology 3-spheres has a perturbative expansion a la Ohtsuki. The presentation of the theory of quantum 3-manifold is self-contained.

26 pages

References in corpus (2)

• The universal Vassiliev-Kontsevich invariant for framed oriented links
• Witten's Invariants of Rational Homology Spheres at Prime Values of $K$ and Trivial Connection Contribution

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• Quantum Groups at Roots of Unity and Modularity
• Cyclotomic completions of polynomial rings
• Spin Modular Categories
• Modular categories of types B,C and D
• Unified quantum invariants and their refinements for homology 3-spheres with 2-torsion
• On modular group representations associated to SO$(p{)}_2$-TQFTs