  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.
