Factorization of Dirac operators on toric noncommutative manifolds

Jens Kaad, Walter D. van Suijlekom*

*Kontaktforfatter for dette arbejde

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Resumé

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.

OriginalsprogEngelsk
TidsskriftJournal of Geometry and Physics
Vol/bind132
Sider (fra-til)282-300
ISSN0393-0440
DOI
StatusUdgivet - 2018

Fingeraftryk

Dirac Operator
factorization
KK-theory
Factorization
operators
Tensor
tensors
Curvature
Torus Action
Factorise
Orbit Space
curvature
Quadrant
orbits
K-theory
Term
Obstruction
quadrants
strata
Orbit

Citer dette

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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 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.",
keywords = "Dirac operators, Toric noncommutative manifolds, Unbounded KK-theory",
author = "Jens Kaad and {van Suijlekom}, {Walter D.}",
year = "2018",
doi = "10.1016/j.geomphys.2018.05.027",
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Factorization of Dirac operators on toric noncommutative manifolds. / Kaad, Jens; van Suijlekom, Walter D.

I: Journal of Geometry and Physics, Bind 132, 2018, s. 282-300.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - Factorization of Dirac operators on toric noncommutative manifolds

AU - Kaad, Jens

AU - van Suijlekom, Walter D.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Dirac operators

KW - Toric noncommutative manifolds

KW - Unbounded KK-theory

U2 - 10.1016/j.geomphys.2018.05.027

DO - 10.1016/j.geomphys.2018.05.027

M3 - Journal article

AN - SCOPUS:85049747854

VL - 132

SP - 282

EP - 300

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -