On Certain Spectral Invariants of Dirac Operators on Noncommutative Tori

Abstract

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 ${\zeta }_{|D|}^{\prime }(0)$ 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.

arXiv comment30 pages

References in corpus (7)

• Type III and spectral triples
• The Gauss-Bonnet Theorem for the noncommutative two torus
• Curved noncommutative torus and Gauss--Bonnet
• An Asymmetric Noncommutative Torus
• The Ricci flow on noncommutative two-tori
• A Riemann-Roch theorem for the noncommutative two torus
• The canonical trace and the noncommutative residue on the noncommutative torus

Cited by in corpus (5)

• A Scalar Curvature Formula For the Noncommutative 3-Torus
• Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals
• Gauss-Bonnet for matrix conformally rescaled Dirac
• Pseudodifferential calculus on noncommutative tori, II. Main properties
• Zeta-regularization and the heat-trace on some compact quantum semigroups