Noncommutative geometry, inspired by quantum physics, describes singular spaces by their noncommutative coordinate algebras and metric structures by Dirac-like operators. Such metric geometries are described mathematically by Connes' theory of spectral triples. These lectures, delivered at an EMS Summer School on noncommutative geometry and its applications, provide an overview of spectral triples based on examples. This introduction is aimed at graduate students of both mathematics and theoretical physics. It deals with Dirac operators on spin manifolds, noncommutative tori, Moyal quantization and tangent groupoids, action functionals, and isospectral deformations. The structural framework is the concept of a noncommutative spin geometry; the conditions on spectral triples which determine this concept are developed in detail. The emphasis throughout is on gaining understanding by computing the details of specific examples. The book provides a middle ground between a comprehensive text and a narrowly focused research monograph. It is intended for self-study, enabling the reader to gain access to the essentials of noncommutative geometry. New features since the original course are an expanded bibliography and a survey of more recent examples and applications of spectral triples. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Author(s): Joseph C. Varilly
Series: EMS Series of Lectures in Mathematics
Publisher: European Mathematical Society
Year: 2006
Language: English
Pages: 122
Introduction......Page 6
Commutative geometry from the noncommutative point of view......Page 10
The Gelfand–Naımark cofunctors......Page 11
The functor......Page 12
Hermitian metrics and spinc structures......Page 14
The Dirac operator and the distance formula......Page 17
Line bundles and the spinor bundle......Page 20
The Dirac operator on the sphere S2......Page 22
Spinor harmonics and the spectrum of D-11mu/......Page 24
Twisted spinor modules......Page 26
A reducible spectral triple......Page 28
The data set......Page 30
Infinitesimals and dimension......Page 32
Smoothness of the algebra......Page 34
Hochschild cycles and orientation......Page 35
Finiteness of the K-cycle......Page 36
Poincaré duality and K-theory......Page 37
The real structure......Page 39
Algebras of Weyl operators......Page 41
The algebra of the noncommutative torus......Page 43
The skeleton of the noncommutative torus......Page 45
A family of spin geometries on the torus......Page 47
The Dixmier trace on infinitesimals......Page 52
Pseudodifferential operators......Page 55
The Wodzicki residue......Page 57
The trace theorem......Page 58
Integrals and zeta residues......Page 60
Moyal quantizers and the Moyal deformation......Page 62
Smooth groupoids......Page 65
The tangent groupoid......Page 67
Moyal quantization as a continuity condition......Page 69
The hexagon and the analytical index......Page 71
Quantization and the index theorem......Page 72
Unitary equivalence of spin geometries......Page 74
Morita equivalence and connections......Page 76
Vector bundles over noncommutative tori......Page 79
Morita-equivalent toral geometries......Page 81
Gauge potentials......Page 83
Algebra automorphisms and the metric......Page 84
The fermionic action......Page 85
The spectral action principle......Page 87
Spectral densities and asymptotics......Page 88
Noncommutative field theories......Page 94
Isospectral deformations......Page 95
Geometries with quantum group symmetry......Page 98
Other developments......Page 102
Bibliography......Page 106
Index......Page 118
