A Chern-Simons action for noncommutative spaces in general with the example SU_q(2)

TL;DR

This paper extends the Chern-Simons action to 3D noncommutative spaces using spectral triples, proves gauge invariance, and explores its properties through the example of the quantum group SU_q(2).

Contribution

It introduces a new formulation of the Chern-Simons action for noncommutative geometries based on spectral triples, differing from previous cyclic cohomology approaches.

Findings

The new action is gauge invariant.

It reduces to the classical action on 3D spin manifolds.

The action for SU_q(2) is non-topological.

Abstract

Witten constructed a topological quantum field theory with the Chern-Simons action as Lagrangian. We define a Chern-Simons action for 3-dimensional spectral triples. We prove gauge invariance of the Chern-Simons action, and we prove that it concurs with the classical one in the case the spectral triple comes from a 3-dimensional spin manifold. In contrast to the classical Chern-Simons action, or a noncommutative generalization of it introduced by A. H. Chamseddine, A. Connes, and M. Marcolli by use of cyclic cohomology, the formula of our definition contains a linear term which shifts the critical points of the action, i. e. the solutions of the corresponding variational problem. Additionally, we investigate and compute the action for a particular example: the quantum group SU_q(2). Two different spectral triples were constructed for SU_q(2). We investigate the Chern-Simons action, defined in the present paper, in both cases, and conclude the non-topological nature of the action. Using the Chern-Simons action as Lagrangian we define and compute the path integral, at least conceptually.