Matrix Model for Discretized Moduli Space

Abstract

We study the algebraic geometrical background of the Penner--Kontsevich matrix model with the potential $Nα\tr {\bigl(- \fr 12 ŁXŁX +\log (1-X)+X\bigr)}$. We show that this model describes intersection indices of linear bundles on the discretized moduli space right in the same fashion as the Kontsevich model is related to intersection indices (cohomological classes) on the Riemann surfaces of arbitrary genera. The special role of the logarithmic potential originated from the Penner matrix model is demonstrated. The boundary effects which was unessential in the case of the Kontsevich model are now relevant, and intersection indices on the discretized moduli space of genus $g$ are expressed through Kontsevich's indices of the genus $g$ and of the lower genera.