Quantum Hyperbolic Invariants Of 3-Manifolds With PSL(2,C)-Characters

We construct {\it quantum hyperbolic invariants} (QHI) for triples $(W,L,\rho )$, where $W$ is a compact closed oriented 3-manifold, $\rho$ is a flat principal bundle over $W$ with structural group $PSL(2,\text{mc})$, and $L$ is a non-empty link in $W$. These invariants are based on the Faddeev-Kashaev’s {\it 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 {\it idealization} of any fixed {\it $\text{Dd}$-triangulation}; the $\text{Dd}$-triangulations are simplicial 1-cocycle descriptions of $(W,\rho )$ in which the link is realized as a Hamiltonian subcomplex. We also discuss how to set the Volume Conjecture for the coloured Jones invariants $J_N(L)$ of hyperbolic knots $L$ in $S^3$ in the framework of the general QHI theory.