Factorization of Dirac operators on toric noncommutative manifolds

[摘要] 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. (C) 2018 Elsevier B.V. All rights reserved.

[发布日期] 2018-10-01

[效力级别]  Proceedings Paper

[关键词] Toric noncommutative manifolds;Dirac operators;Unbounded KK-theory