We construct a new family, indexed by odd integers
, of
–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
–bordisms supported by
compact oriented –manifolds
with a properly embedded
framed tangle and an
arbitrary –character
of
(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
with marked boundary components a tensor built on the matrix
dilogarithms, which is holomorphic in the boundary parameters. When
the QHFT
tensors are scalar-valued, and coincide with the Cheeger–Chern–Simons invariants of
–characters
on closed manifolds or cusped hyperbolic manifolds. We establish surgery formulas
for QHFT partitions functions and describe their relations with the quantumhyperbolic invariants of Baseilhac and Benedetti (either defined for unframed links in
closed manifolds and characters trivial at the link meridians, or cusped hyperbolic
–manifolds). For
every –character
of a punctured surface, we produce new families of conjugacy classes of “moderately
projective" representations of the mapping class groups.
Keywords
hyperbolic geometry, quantum field theory, mapping class
group representations, quantum invariants,
Cheeger–Chern–Simons class, dilogarithms
