Résumé
We construct a new family, indexed by odd integers N >= 1, of (2 + 1)-dimensional quantum field theories that we call quantum hyperbolicfield theories (QBFT), and we study its main structural properties. The QHFT are defined for marked (2 + 1)-bordisms supported by compact oriented 3-manifolds Y with a properly embedded framed tangle L-F and an arbitrary PSL(2, C)-character rho of Y \ L-F (covering, for example, the case of hyperbolic cone manifolds). The marking of QHFT bordisms includes a specific set of parameters for the space of pleated hyperbolic structures on punctured surfaces. Each QHFT associates in a constructive way to any triple (Y, L-F, rho) with marked boundary components a tensor built on the matrix dilogarithms, which is holomorphic in the boundary parameters. When N = 1 the QHFT tensors; are scalar-valued, and coincide with the Cheeger-Chem-Simons invariants of PSL (2, C)-characters on closed manifolds or cusped hyperbolic manifolds. We establish surgery formulas for QHFT partitions functions and describe their relations with the quantum hyperbolic invariants of Baseilhac and Benedetti [3; 4] (either defined for unframed links in closed manifolds and characters trivial at the link meridians, or cusped hyperbolic 3-manifolds). For every PSL (2, C) -character of a punctured surface, we produce new families of conjugacy classes of "moderately projective" representations of the mapping class groups.