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Contents
Contents
Dirac Operators and Spectral Geometry
Joseph C. Várilly
January 2006
Contents
Introduction and Overview
Clifford algebras and spinor representations
Clifford algebras
The universality property
The trace
Periodicity
Chirality
and
groups
The Lie algebra of
Orthogonal complex structures
Irreducible representations of
Representations of
Spinor modules over compact Riemannian manifolds
Remarks on Riemannian geometry
Clifford algebra bundles
The existence of
structures
Morita equivalence for (commutative) unital algebras
Classification of spinor modules
The spin connection
Epilogue: counting the spin structures
Dirac operators
The metric distance property
Symmetry of the Dirac operator
Selfadjointness of the Dirac operator
The Schrödinger-Lichnerowicz formula
The spectral growth of the Dirac operator
Spectral Growth and Dixmier Traces
Definition of spectral triples
Logarithmic divergence of spectra
Some eigenvalue inequalities
Dixmier traces
Symbols and Traces
Classical pseudodifferential operators
Homogeneity of distributions
The Wodzicki residue
Dixmier trace and Wodzicki residue
Spectral Triples: General Theory
The Dixmier trace revisited
Regularity of spectral triples
Pre-C*-algebras
Real spectral triples
Summability of spectral triples
Spectral Triples: Examples
Geometric conditions on spectral triples
Isospectral deformations of commutative spectral triples
The Moyal plane as a nonunital spectral triple
A geometric spectral triple over
Exercises
Examples of Dirac operators
The circle
The (flat) torus
The Hodge-Dirac operator on
The Dirac operator on the sphere
The spinor bundle
on
The spin connection
over
Spinor harmonics and the Dirac operator spectrum
Spin
Dirac operators on the 2-sphere
A spectral triple on the noncommutative torus
Bibliography
About this document ...
Pawel Witkowski 2006-03-14
and 
groups
structures
Dirac operators on the 2-sphere
on 
over