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Noncommutative Circle Bundles and New Dirac Operators

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Abstract

We study spectral triples over noncommutative principal U(1) bundles. Basing on the classical situation and the abstract algebraic approach, we propose an operatorial definition for a connection and compatibility between the connection and the Dirac operator on the total space and on the base space of the bundle. We analyze in details the example of the noncommutative three-torus viewed as a U(1) bundle over the noncommutative two-torus and find all connections compatible with an admissible Dirac operator. Conversely, we find a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection.

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Authors and Affiliations

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

    Ludwik Dąbrowski

  2. Institute of Physics, Jagiellonian University, Reymonta 4, 30-059, Kraków, Poland

    Andrzej Sitarz

  3. Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, Warszawa, 00-950, Poland

    Andrzej Sitarz

Authors
  1. Ludwik Dąbrowski
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  2. Andrzej Sitarz
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Corresponding author

Correspondence to Ludwik Dąbrowski.

Additional information

Communicated by A. Connes

Partially supported by GSQS 230836 (IRSES, EU) and PRIN 2010-2012 (MIUR, Italy).

Partially supported by MNII grants 189/6.PRUE/2007/7 and N 201 1770 33.

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Dąbrowski, L., Sitarz, A. Noncommutative Circle Bundles and New Dirac Operators. Commun. Math. Phys. 318, 111–130 (2013). https://doi.org/10.1007/s00220-012-1550-8

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  • DOI: https://doi.org/10.1007/s00220-012-1550-8

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