The Chern-Simons Action in Non-Commutative Geometry

TL;DR

This paper defines Chern-Simons actions within non-commutative geometry, illustrating their relation to topological gravity on certain product spaces, and demonstrates how to derive dynamical actions from these topological constructs.

Contribution

It introduces a general framework for Chern-Simons actions in non-commutative geometry and connects them to topological gravity on specific non-commutative spaces.

Findings

Chern-Simons actions correspond to topological gravity actions on product spaces.

Explicit examples illustrate the construction of these actions.

A method to extract dynamical actions from topological Chern-Simons actions.

Abstract

A general definition of Chern-Simons actions in non-commutative geometry is proposed and illustrated in several examples. These 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 non-commutative 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 contrasting the space of field configurations in these examples in an appropriate manner one is able to extract dynamical actions from Chern-Simons actions.