Factorization of Dirac operators on toric noncommutative manifolds

TL;DR

This paper demonstrates a factorization of the Dirac operator on the Connes-Landi 4-sphere within unbounded KK-theory, revealing a tensor sum decomposition and its relation to the interior Kasparov product, with generalizations to other toric noncommutative manifolds.

Contribution

It introduces a novel factorization of Dirac operators on toric noncommutative manifolds using unbounded KK-theory, including a curvature obstruction term.

Findings

Tensor sum of Dirac operators matches the Dirac operator on the sphere

The tensor sum represents the interior Kasparov product in bivariant K-theory

Curvature term obstructs tensor sum decomposition in unbounded KK-theory

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.