Research ArticlesNo Access

DIRAC OPERATORS ON NONCOMMUTATIVE PRINCIPAL CIRCLE BUNDLES

LUDWIK DABROWSKI

SISSA (Scuola Internazionale Superiore di Studi Avanzati), via Bonomea 265, 34136 Trieste, Italy

Search for more papers by this author

ANDRZEJ SITARZ

Institute of Physics, Jagiellonian University, Reymonta 4, Cracow 30-059, Poland

Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, Warsaw 00-950, Poland

Search for more papers by this author

, and

ALESSANDRO ZUCCA

SISSA (Scuola Internazionale Superiore di Studi Avanzati), via Bonomea 265, 34136 Trieste, Italy

Search for more papers by this author

https://doi.org/10.1142/S0219887814500121Cited by:6 (Source: Crossref)

Abstract

We study spectral triples over noncommutative principal U(1)-bundles of arbitrary dimension and a compatibility condition between the connection and the Dirac operator on the total space and on the base space of the bundle. Examples of low-dimensional noncommutative tori are analyzed in more detail and all connections found that are compatible with an admissible Dirac operator. Conversely, a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection is exhibited. These examples are extended to the theta-deformed principal U(1)-bundle .

Keywords:

AMSC: 58B34, 55R10, 16W30

References

B. Ammann, Sém. Th. Spec. Géom Inst. Fourier Grenoble 16, 33 (1998). Google Scholar
B. Ammann and Ch. Bär, Ann. Global Anal. Geom. 16, 221 (1998). Web of Science, Google Scholar
T. Brzezinski and S. A. Fairfax, Axioms 1(2), 201 (2012). Google Scholar
T. Brzezinski and S. Majid, Quantum group gauge theory on quantum spaces, Commun. Math. Phys. 157 (1993) 591; Erratum: 167 (1995) 235 . Google Scholar
T. Brzezinski and A. Sitarz, Group 24. Physical and Mathematical Aspects of Symmetries, eds. J.-P. Gazeauet al. (IOP, 2003) pp. 791–794. Google Scholar
T. Brzezinski and B. Zielinski, J. Geom. Phys. 62, 1097 (2012). Web of Science, Google Scholar
A. Connes , Noncommutative Geometry ( Academic Press , San Diego , 1994 ) . Google Scholar
A. Connes, Lett. Math. Phys. 34, 203 (1995). Web of Science, Google Scholar
A. Connes, Commun. Math. Phys. 182, 155 (1996). Web of Science, Google Scholar
A. Connes and M. Dubois-Violette, Commun. Math. Phys. 230, 539 (2002). Web of Science, Google Scholar
A. Connes and G. Landi, Commun. Math. Phys. 221, 141 (2001). Web of Science, Google Scholar
L. Dabrowski and G. Dossena, Int. J. Geom. Meth. Mod. Phys. 8, 1833 (2011). Link, Web of Science, Google Scholar
L. Dabrowski, H. Grosse and P. M. Hajac, Commun. Math. Phys. 220, 301 (2001). Web of Science, Google Scholar
L. Dabrowski and A. Sitarz, Commun. Math. Phys. 318, 111 (2013). Web of Science, Google Scholar
L. Dabrowski and A. Sitarz, J. Math. Phys. 54, 013518 (2013). Web of Science, Google Scholar
J. M. Gracia-Bondia , J. C. Várilly and H. Figueroa , Elements of Noncommutative Geometry ( Birkhäuser , Boston , 2001 ) . Google Scholar
P. M. Hajac, Commun. Math. Phys. 182, 579 (1996). Web of Science, Google Scholar
P. M. Hajac and S. Majid, Commun. Math. Phys. 206, 247 (1999). Web of Science, Google Scholar
M. Rieffel, Mem. Amer. Math. Soc. 106, 506 (1993). Google Scholar
A. Sitarz, Noncommutative Geometry and Quantum Groups 61, eds. P. M. Hajac and W. Pusz (Banach Center Publications, Warszawa, 2003) pp. 231–263. Google Scholar