THE CHERN-SIMONS ACTION IN NONCOMMUTATIVE GEOMETRY

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