Noncommutative Chern-Simons theory on the quantum 3-sphere $S^3_\theta$

Abstract

We consider the $\theta$-deformed quantum three sphere $S^3_\theta$ and study its Chern--Simons theory from a spectral point of view. We first construct a spectral triple on $S^3_\theta$ as a generalization of the Dirac geometry on $S^3 $. Since the choice of Dirac operator is not unique, we give two more natural spectral triples on $S^3_\theta$ related to the standard round metric. We then compute the Chern--Simons action with respect to the three spectral triples, it turns out that it is not a topological invariant, that is, it depends on the choice of Dirac operators.