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2012

Curved noncommutative torus and Gauss--Bonnet

Dabrowski, Ludwik, Sitarz, Andrzej

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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.

Curved noncommutative torus and Gauss--Bonnet · Around

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