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Dirac operators for coadjoint orbits of compact Lie groups

The coadjoint orbits of compact Lie groups carry many Kähler structures, which include a Riemannian metric and a complex structure. We provide a fairly explicit formula for the Levi-Civita connection of the Riemannian metric, and we use the complex structure to give a fairly explicit construction of a canonical Dirac operator for the Riemannian metric, in a way that avoids use of the spinc groups. Substantial parts of our results apply to compact almost-Hermitian homogeneous spaces, and to other connections besides the Levi-Civita connection. For these other connections we give a criterion that is both necessary and sufficient for their Dirac operator to be formally self-adjoint. We hope to use the detailed results given here to clarify statements in the literature of high-eneregy physics concerning “Dirac operators” for matrix algebras that converge to coadjoint orbits. To facilitate this we employ here only global methods—we never use local coordinate charts, and we use the cross-section modules of vector bundles.

Titel: Dirac operators for coadjoint orbits of compact Lie groups
Verfasser: Rieffel, Marc A. GND
Organisation: FB 10: Mathematik und Informatik
Dokumenttyp: Artikel
Medientyp: Text
Erscheinungsdatum: 2009
Publikation in MIAMI: 20.08.2009
Datum der letzten Änderung: 11.05.2015
Zeitschrift/Periodikum: Münster Journal of Mathematics / 2 (2009), S. 265 - 298
Fachgebiete: Mathematik
Sprache: Englisch
Format: PDF-Dokument
URN: urn:nbn:de:hbz:6-10569448769