Matrix Model for Discretized Moduli Space

TL;DR

This paper explores the algebraic geometry underlying a Penner--Kontsevich matrix model, linking it to intersection indices on discretized moduli spaces and highlighting the significance of boundary effects and the logarithmic potential.

Contribution

It establishes a connection between the Penner--Kontsevich matrix model and intersection indices on discretized moduli spaces, extending the understanding of boundary effects and the role of the logarithmic potential.

Findings

Model describes intersection indices on discretized moduli space.

Boundary effects are significant in the discretized setting.

Intersection indices relate to Kontsevich's indices across genera.

Abstract

We study the algebraic geometrical background of the Penner--Kontsevich matrix model with the potential $N\alpha \tr {\bigl(- \fr 12 \L X\L X +\log (1-X)+X\bigr)}$. 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.