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2014

An Asymmetric Noncommutative Torus

Dabrowski, Ludwik, Sitarz, Andrzej

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We introduce a family of spectral triples that describe the curved noncommutative two-torus.

The relevant family of new Dirac operators is given by rescaling one of two terms in the flat Dirac operator.
We compute the dressed scalar curvature and show that the Gauss-Bonnet theorem holds (which is not covered by the general result of Connes and Moscovici).

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Cohen P.B., Connes A., Conformal geometry of the irrational rotation algebra, Preprint MPI (92-93), 1992
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Earlier work this paper cites.
Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994
1994
Earlier work this paper cites.
Connes A., Moscovici H., The local index formula in noncommutative geometry,
1995
Earlier work this paper cites.
Connes A., Moscovici H., Type III and spectral triples, in Traces in Number Theory, Geometry and Quantum Fields,
2008
Earlier work this paper cites.
Connes A., Tretkoff P., The Gauss–Bonnet theorem for the noncommutative two torus, in Noncommutative Geometry, Arithmetic, and Related Topics, Johns Hopkins Univ. Press, Baltimore, MD, 2011, 141–158,
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2011
Earlier work this paper cites.

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Then

Fathizadeh F., Khalkhali M., Scalar curvature for the noncommutative two torus,
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2013
Later among the works it cites.
Fathizadeh F., Khalkhali M., Weyl’s law and Connes’ trace theorem for noncommutative two tori,
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2013
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Rosenberg J., Levi-Civita’s theorem for noncommutative tori,
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2013
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Connes A., Moscovici H., Modular curvature for noncommutative two-tori,
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2014
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Ponge R., Wang H., Noncommutative geometry and conformal geometry. III. Vafa–Witten inequality and Poincaré duality,
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Lesch M., Divided differences in noncommutative geometry: rearrangement lemma, functional calculus and expansional formula,

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2015

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