Factorization of Dirac operators on toric noncommutative manifolds

Jens Kaad, Walter D. van Suijlekom*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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We factorize the Dirac operator on the Connes–Landi 4-sphere in unbounded KK-theory. We show that a family of Dirac operators along the orbits of the torus action defines an unbounded Kasparov module, while the Dirac operator on the principal orbit space –an open quadrant in the 2-sphere – defines a half-closed chain. We show that the tensor sum of these two operators coincides up to unitary equivalence with the Dirac operator on the Connes–Landi sphere and prove that this tensor sum is an unbounded representative of the internal Kasparov product in bivariant K-theory. We also generalize our results to Dirac operators on all toric noncommutative manifolds subject to a condition on the principal stratum. We find that there is a curvature term that arises as an obstruction for having a tensor sum decomposition in unbounded KK-theory. This curvature term can however not be detected at the level of bounded KK-theory.

Original languageEnglish
JournalJournal of Geometry and Physics
Pages (from-to)282-300
Publication statusPublished - 2018


  • Dirac operators
  • Toric noncommutative manifolds
  • Unbounded KK-theory


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