Dirac operators for coadjoint orbits of compact Lie groups

Abstract

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 Spin^c 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.

34 pages; v2: many local improvements, including many reflecting suggestions of the referee, including some important ones

References in corpus (2)

• Leibniz seminorms for "Matrix algebras converge to the sphere"
• Ginsparg-Wilson Dirac operator in the monopole backgrounds on the fuzzy 2-sphere

Cited by in corpus (6)

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• Geometry and topology of CC and CQ states
• Dirac operators for matrix algebras converging to coadjoint orbits
• Twist star products and Morita equivalence
• On the Dolbeault--Dirac Operator of Quantized Symmetric Spaces
• Quantum Algebras Associated to Irreducible Generalized Flag Manifolds