THE CHERN-SIMONS ACTION IN NONCOMMUTATIVE GEOMETRY

A general definition of Chern-Simons actions in noncommutative geometr y is proposed and illustrated in several examples. These examples are based on ''space-times'' which are products of even-dimensional, Riema nnian spin manifolds by a discrete (two-point) set. If the algebras of operators describing the noncommutative spaces are generated by fun ctions over such ''space-times'' with values in certain Clifford algeb ras the Chern-Simons actions turn out to be the actions of topological gravity on the even-dimensional spin manifolds. By constraining the s pace of field configurations in these examples in an appropriate manne r one is able to extract dynamical actions from Chern-Simons actions.