Quantum hyperbolic invariants of 3-manifolds with PSL(2,C)-characters

We construct quantum hyperbolic invariants (QHI) for triples (W,L,ρ), where W is a compact closed oriented 3-manifold, ρ is a flat principal bundle over W with structural group PSL(2,C), and L is a non-empty link in W. These invariants are based on the Faddeev–Kashaevʹs quantum dilogarithms, interpreted as matrix-valued functions of suitably decorated hyperbolic ideal tetrahedra. They are explicitly computed as state sums over the decorated hyperbolic ideal tetrahedra of the idealization of any fixed D-triangulation; the D-triangulations are simplicial 1-cocycle descriptions of (W,ρ) in which the link is realized as a Hamiltonian subcomplex. We also discuss how to set the Volume Conjecture for the coloured Jones invariants JN(L) of hyperbolic knots L in S3 in the framework of the general QHI theory.