Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 075, 11 pages      arXiv:1406.4645      https://doi.org/10.3842/SIGMA.2015.075

An Asymmetric Noncommutative Torus

Ludwik Dąbrowski a and Andrzej Sitarz bc
a) SISSA (Scuola Internazionale Superiore di Studi Avanzati), via Bonomea 265, 34136 Trieste, Italy
b) Institute of Physics, Jagiellonian University, Stanisława Łojasiewicza 11, 30-348 Kraków, Poland
c) Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland

Received December 09, 2014, in final form September 17, 2015; Published online September 26, 2015

Abstract
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).

Key words: noncommutative geometry; Gauss-Bonnet; spectral triple.

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