Moduli Spaces of Curves with Homology Chains and c=1 Matrix Models

TL;DR

This paper introduces a new matrix model that generalizes Penner-Kontsevich models to include homology chains on curves, providing a novel representation of $c=1$ matter coupled to 2D quantum gravity with a circular target space.

Contribution

It extends existing models by incorporating homology chains and kinetic terms, leading to a new matrix model for $c=1$ matter coupled to 2D quantum gravity.

Findings

Realizes a simple dynamics of $ ext{Z}_k$-chains on surfaces.

Provides a matrix model representation of $c=1$ matter coupled to 2D quantum gravity.

Connects to models studied by Gross and Klebanov.

Abstract

We show that introducing a periodic time coordinate in the models of Penner-Kontsevich type generalizes the corresponding constructions to the case of the moduli space ${\cal S}_{gn}^k$ of curves $C$ with homology chains $\gamma\in H_1(C,\zet_k)$. We make a minimal extension of the resulting models by adding a kinetic term, and we get a new matrix model which realizes a simple dynamics of $\zet_k$-chains on surfaces. This gives a representation of $c=1$ matter coupled to two-dimensional quantum gravity with the target space being a circle of finite radius, as studied by Gross and Klebanov.