Invertible cohomological field theories and Weil-Petersson volumes

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 fr om Witten's free energy by a change of variables given by Schur polynomials . Since this generating function has a natural extension to the moduli spac e of invertible Cohomological Field Theories, this suggests the existence o f a "very large phase space", correlation functions on which include Wedge integrals studied by C. Faber and R. Pandharipande. From this formula we de rive an asymptotical expression for the Weil-Petersson volume as conjecture d 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 groupoi d.