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arXiv · 1803.08921

Factorization of Dirac operators on toric noncommutative manifolds

Jens Kaad·Walter D. van Suijlekom
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Abstract

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 interior 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.

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BibTeXRIS

Jens Kaad, Walter D. van Suijlekom. 2018-03-23. Factorization of Dirac operators on toric noncommutative manifolds. https://doi.org/10.1016/j.geomphys.2018.05.027

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