  We show that the generating function for the higher Weil-Petersson volumes of
the moduli spaces of stable curves with marked points can be obtained from
Witten's free energy by a change of variables given by Schur polynomials. Since
this generating function has a natural extension to the moduli space of
invertible Cohomological Field Theories, this suggests the existence of a
``very large phase space'', correlation functions on which include Hodge
integrals studied by C. Faber and R. Pandharipande. From this formula we derive
an asymptotical expression for the Weil-Petersson volume as conjectured by C.
Itzykson. We also discuss a topological interpretation of the genus expansion
formula of Itzykson-Zuber, as well as a related bialgebra acting upon quantum
cohomology as a complex version of the classical path groupoid.
