Matrix Models and Geometry of Moduli Spaces

We give the description of discretized moduli spaces (d.m.s.) M disc g,n introduced in [1] in terms of discrete de Rham cohomologies for moduli spaces Mg,n. The generating function for intersection indices (cohomological classes) of d.m.s. is found. Classes of highest degree coincide with the ones for the continuum moduli space Mg,n. To show it we use a matrix model technique. The Kontsevich matrix model is the generating function in the continuum case, and the matrix model with the potential Nα tr ( − 1 4 ΛXΛX − 1 2 log(1−X)− 1 2 X ) is the one for d.m.s. In the latest case the effects of Deligne–Mumford reductions become relevant, and we use the stratification procedure in order to express integrals over open spaces M g,n in terms of intersection indices, which are to be calculated on compactified spaces M disc g,n . We find and solve constraint equations on partition function Z of our matrix model expressed in times for d.m.s.: tm = tr ∂ ∂λm 1 e λ−1 . It appears that Z depends only on even times and Z[t· ] = C(αN) e A e F ({t − 2n })+F ({−t 2n , where F ({t±2n}) is a logarithm of the partition function of the Kontsevich model, A being a quadratic differential operator in ∂ ∂t ± 2n . This work was supported by RFFI grant N 94–01–00285. ∗E–mail: [email protected]