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2010

Noncommutative circle bundles and new Dirac operators

Dabrowski, Ludwik, Sitarz, Andrzej

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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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A.Connes, Gravity coupled with matter and foundation of non-commutative geometry
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P.Gilkey, J.Leahy, J.H. Park, The spectral geometry of the Hopf fibration
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P.M.Hajac, Strong connections on quantum principal bundles
1996
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