Conformal modes in simplicial quantum gravity and the Weil-Petersson volume of moduli space

Our goal here is to present a detailed analysis connecting the anomalous scaling properties of 2D simplicial quantum gravity to the geometry of the moduli space M_{g,N} of genus g Riemann surfaces with N punctures. In the case of pure gravity we prove that the scaling properties of the set of dynamical triangulations with N vertices are directly provided by the large N asymptotics of the Weil-Petersson volume of M_{g,N}, recently discussed by Manin and Zograf. Such a geometrical characterization explains why dynamical triangulations automatically take into account the anomalous scaling properties of Liouville theory. In the case of coupling with conformal matter we briefly argue that the anomalous scaling of the resulting discretized theory should be related to the Gromov-Witten invariants of the moduli space M_{g,N}(X,\beta) of stable maps from (punctured Riemann surfaces associated with) dynamical triangulations to a manifold X parameterizing the conformal matter configurations.