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We construct a new family, indexed by odd integers N≥1, of
(2+1)–dimensional quantum field theories that we call quantum
hyperbolic field theories (QHFT), 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 LF and an arbitrary
PSL(2,C)–character ρ of Y╲LF
(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,LF,ρ)
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–Chern–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
(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.
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