MODULI SPACES OF CURVES WITH HOMOLOGY CHAINS AND C=1 MATRIX MODELS

We show that introducing a periodic time coordinate in the models of P enner-Kontsevich type generalizes the corresponding constructions to t he case of the moduli space S(g,n)k of curves C with homology chains g amma is-an-element-of H-1 (C, Z(k)). We make a minimal extension of th e resulting models by adding a kinetic term, and we get a new matrix m odel which realizes a simple dynamics of Z(k)-chains on surfaces. This gives a representation of c = 1 matter coupled to two-dimensional qua ntam gravity with the target space being a circle of finite radius, as studied by Gross and Klebanov.