Curved noncommutative torus and Gauss--Bonnet

Details Curved noncommutative torus and Gauss--Bonnet Curved noncommutative torus and Gauss--Bonnet

2012-04-02

arxiv.org/abs/1204.0420v1

Mathematics

Quantum Algebra

13 pages, LaTeX

Scientific paper

We study perturbations of the flat geometry of the noncommutative two-dimensional torus T^2_\theta (with irrational \theta). They are described by spectral triples (A_\theta, \H, D), with the Dirac operator D, which is a differential operator with coefficients in the commutant of the (smooth) algebra A_\theta of T_\theta. We show, up to the second order in perturbation, that the zeta-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.

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