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 SUq(2). Two different spectral triples were constructed for SUq(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.
Cite this article
Oliver Pfante, A Chern–Simons action for noncommutative spaces in general with the example SU<sub><var>q</var></sub>(2). J. Noncommut. Geom. 8 (2014), no. 3, pp. 611–654DOI 10.4171/JNCG/166