  We use quantum invariants to define an analytic family of representations for
the mapping class group of a punctured surface. The representations depend on a
complex number A with |A| <= 1 and act on an infinite-dimensional Hilbert
space. They are unitary when A is real or imaginary, bounded when |A|<1, and
only densely defined when |A| = 1 and A is not a root of unity. When A is a
root of unity distinct from 1, -1, i, -i the representations are
finite-dimensional and isomorphic to the "Hom" version of the well-known TQFT
quantum representations.
  The unitary representations in the interval [-1,0] interpolate analytically
between two natural geometric unitary representations, the SU(2)-character
variety representation studied by Goldman and the multicurve representation
induced by the action of the mapping class group on multicurves.
  The finite-dimensional representations converge analytically to the
infinite-dimensional ones. We recover Marche and Narimannejad's convergence
theorem, and Andersen, Freedman, Walker and Wang's asymptotic faithfulness,
that states that the image of a non-central mapping class is always non-trivial
after some level r. When the mapping class is pseudo-Anosov we give a simple
polynomial estimate of the level r in term of its dilatation.
