A Chern–Simons action for noncommutative spaces in general with the example ${\mathrm{SU}}_q(2)$

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 ${\mathrm{SU}}_q(2)$. Two different spectral triples were constructed for ${\mathrm{SU}}_q(2)$). 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.