On Certain Spectral Invariants of Dirac Operators on Noncommutative Tori

The spectral eta function for certain families of Dirac operators on noncommutative $3$-torus is considered and the regularity at zero is proved. By using variational techniques, we show that $\eta_{D}(0)$ is a conformal invariant. By studying the Laurent expansion at zero of $\text{TR} (|D|^{-z})$, the conformal invariance of $\zeta'_{|D|}(0)$ for noncommutative $3$-torus is proved. Finally, for the coupled Dirac operator, a local formula for the variation $\partial_A\eta_{D+A}(0)$ is derived which is the analogue of the so called induced Chern-Simons term in quantum field theory literature.