The Chern–Simons action in noncommutative geometry

Abstract

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.