Chern–Simons theory for the noncommutative three-torus C∞(TΘ3)

In a previous paper we defined a Chern–Simons action for noncommutative spaces, i.e. spectral triples. In the present paper we compute this action explicitly for the noncommutative three-torus C∞(TΘ3), a ∗∗-algebra generated by three unitaries, and its spectral triple constructed by D. Essouabri, B. Iochum, C. Levy, and A. Sitarz. In connection with this computation we calculate the first coefficient in the loop expansion series of the corresponding Feynman path integral with the Chern–Simons action as Lagrangian. The result does not depend on the deformation matrix ΘΘ and is always equal to 0.