# Refined open intersection numbers and the Kontsevich-Penner matrix model

## TL;DR

This paper refines open intersection numbers on moduli spaces of Riemann surfaces with boundary, computes these refined numbers, and proposes a matrix model conjecture linking them to the Kontsevich-Penner model, providing supporting evidence.

## Contribution

It introduces a refinement of open intersection numbers based on boundary components, computes these numbers, and formulates a conjecture relating them to a matrix model.

## Key findings

- Computed all refined open intersection numbers for different boundary counts.
- Constructed a matrix model for the generating series of these numbers.
- Presented evidence supporting the conjectured equivalence to the Kontsevich-Penner matrix model.

## Abstract

A study of the intersection theory on the moduli space of Riemann surfaces with boundary was recently initiated in a work of R. Pandharipande, J. P. Solomon and the third author, where they introduced open intersection numbers in genus 0. Their construction was later generalized to all genera by J. P. Solomon and the third author. In this paper we consider a refinement of the open intersection numbers by distinguishing contributions from surfaces with different numbers of boundary components, and we calculate all these numbers. We then construct a matrix model for the generating series of the refined open intersection numbers and conjecture that it is equivalent to the Kontsevich-Penner matrix model. An evidence for the conjecture is presented. Another refinement of the open intersection numbers, which describes the distribution of the boundary marked points on the boundary components, is also discussed.