We organize the quantum hyperbolic invariants (QHI) of
–manifolds
into sequences of rational functions indexed by the odd integers
and defined on moduli spaces of geometric structures refining
the character varieties. In the case of one-cusped hyperbolic
–manifolds
we generalize the QHI and get rational functions
depending on a finite set
of cohomological data
called
weights. These functions are regular on a determined Abelian covering of degree
of a Zariski open subset, canonically associated to
,
of the geometric component of the variety of augmented
–characters
of
.
New combinatorial ingredients are a weak version of branchings which exists on
every triangulation, and state sums over weakly branched triangulations,
including a sign correction which eventually fixes the sign ambiguity of the
QHI. We describe in detail the invariants of three cusped manifolds, and
present the results of numerical computations showing that the functions
depend on the
weights as
,
and recover the volume for some specific choices of the weights.
Keywords
quantum invariants, 3–manifolds, character varieties,
Chern–Simons theory, volume conjecture