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

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 ${\mathcal{S}}_{gn}^k$ of curves $C$ with homology chains $γ\in H_1(C,\zet_k)$. 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.

IFUM 459/FT (LaTeX, 9 pages; a few misprints have been corrected and the introduction has been slightly modified)