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 P SL(2, C), and L is a non-empty link in W . These invariants are based on the Faddeev-Kashaev's quantum dilogarithms, interpreted as matrix valued functions of suitably decorated hyperbolic ideal tetrahedra. They are explicitely 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 S 3 in the framework of the general QHI theory.

We introduce a family of matrix dilogarithms, which are automorphisms of C N ⊗ C N , N being any odd positive integer, associated to hyperbolic ideal tetrahedra equipped with an additional decoration. The matrix dilogarithms satisfy fundamental five-term identities that correspond to decorated versions of the 2 → 3 move on 3-dimensional triangulations. Together with the decoration, they arise from the solution we give of a symmetrization problem for a specific family of basic matrix dilogarithms, the classical (N = 1) one being the Rogers dilogarithm, which only satisfy one special instance of five-term identity. We use the matrix dilogarithms to construct invariant state sums for closed oriented 3-manifolds W endowed with a flat principal P SL(2, C)-bundle ρ, and a fixed non empty link L if N > 1, and for (possibly "marked") cusped hyperbolic 3-manifolds M . When N = 1 the state sums recover known simplicial formulas for the volume and the Chern-Simons invariant. When N > 1, the invariants for M are new; those for triples (W, L, ρ) coincide with the quantum hyperbolic invariants defined in [3], though our present approach clarifies substantially their nature. We analyse the structural coincidences versus discrepancies between the cases N = 1 and N > 1, and we formulate "Volume Conjectures", having geometric motivations, about the asymptotic behaviour of the invariants when N → ∞.