  We introduce a notion of generalized modular functors with Hilbert spaces of
infinite dimension in general, and show that a generalized modular functor with
data of conformal dimensions determines uniquely wave functions as its flat
sections. Furthermore, we study an example of generalized modular functors
derived from the Liouville conformal field theory. In this example, wave
functions are seen to be gluing conformal blocks which are identified with
eigenfunctions in the quantum Teichm\"uller theory.
