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Curved noncommutative torus and Gauss--Bonnet

TitleCurved noncommutative torus and Gauss--Bonnet
Publication TypeJournal Article
Year of Publication2013
AuthorsDabrowski, L, Sitarz, A
JournalJournal of Mathematical Physics. Volume 54, Issue 1, 22 January 2013, Article number 013518

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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