Matrix Models: a Way to Quantum Moduli Spaces

We give the description of discretized moduli spaces (d.m.s.) M g,n introduced in [1] in terms of a discrete de Rham cohomologies for each moduli space Mg,n of a genus g, n being the number of punctures. We demonstrate that intersection indices (cohomological classes) calculated for d.m.s. coincide with the ones for the continuum moduli space Mg,n compactified by Deligne and Mumford procedure. To show it we use a matrix model technique. The Kontsevich matrix model is a generating function for these indices in the continuum case, and the matrix model with the potential Nα tr (−12ΛXΛX + log(1−X) +X) is the one for d.m.s. In the last case the effects of reductions become relevant, but we use the stratification procedure in order to express integrals over open spaces Mdisc g,n in terms of intersection indices which are to be calculated on compactified spaces. The coincidence of the cohomological classes for both continuum and d.m.s. models enables us to propose the existence of a quantum group structure on d.m.s. Then d.m.s. are nothing but cyclic (exceptional) representations of a quantum group related to a moduli space Mg,n. Considering the explicit expressions for integrals of Chern classes over Mg,n and M g,n we conjecture that each moduli space Mg,n in the Kontsevich parametrization can be presented as a coset Mg,n = Td/G, d = 3g − 3 + n, where T d is some d–dimensional complex torus and G is a finite order symmetry group of T d. ∗E–mail: [email protected] and [email protected]