Curved noncommutative torus and Gauss–Bonnet

Abstract

We study perturbations of the flat geometry of the noncommutative two-dimensional torus \documentclass[12pt]{minimal}\begin{document}$\mathbb {T}^2_\theta$\end{document}Tθ2 (with irrational θ). They are described by spectral triples \documentclass[12pt]{minimal}\begin{document}$(A_\theta , \mathcal {H}, D)$\end{document}(Aθ,H,D), with the Dirac operator D, which is a differential operator with coefficients in the commutant of the (smooth) algebra Aθ of \documentclass[12pt]{minimal}\begin{document}$\mathbb {T}_\theta$\end{document}Tθ. We show, up to the second order in perturbation, that the ζ-function at 0 vanishes and so the Gauss-Bonnet theorem holds. We also calculate first two terms of the perturbative expansion of the corresponding local scalar curvature.