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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.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Commutative geometry from the noncommutative point of view 1
1.1 The Gelfand–Na˘ımark cofunctors . . . . . . . . . . . . . . . . . . . 2
1.2 The functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Hermitian metrics and spinc structures . . . . . . . . . . . . . . . . . 5
1.4 The Dirac operator and the distance formula . . . . . . . . . . . . . . 8
2 Spectral triples on the Riemann sphere 11
2.1 Line bundles and the spinor bundle . . . . . . . . . . . . . . . . . . . 11
2.2 The Dirac operator on the sphere S2 . . . . . . . . . . . . . . . . . . 13
2.3 Spinor harmonics and the spectrum of D/ . . . . . . . . . . . . . . . . 15
2.4 Twisted spinor modules . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 A reducible spectral triple . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Real spectral striples: the axiomatic foundation 21
3.1 The data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Infinitesimals and dimension . . . . . . . . . . . . . . . . . . . . . . 23
3.3 The first-order condition . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Smoothness of the algebra . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Hochschild cycles and orientation . . . . . . . . . . . . . . . . . . . 26
3.6 Finiteness of the K-cycle . . . . . . . . . . . . . . . . . . . . . . . . 27
3.7 Poincaré duality and K-theory . . . . . . . . . . . . . . . . . . . . . 28
3.8 The real structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Geometries on the noncommutative torus 32
4.1 Algebras ofWeyl operators . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 The algebra of the noncommutative torus . . . . . . . . . . . . . . . 34
4.3 The skeleton of the noncommutative torus . . . . . . . . . . . . . . . 36
4.4 A family of spin geometries on the torus . . . . . . . . . . . . . . . . 38
5 The noncommutative integral 43
5.1 The Dixmier trace on infinitesimals . . . . . . . . . . . . . . . . . . 43
5.2 Pseudodifferential operators . . . . . . . . . . . . . . . . . . . . . . 46
5.3 TheWodzicki residue . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4 The trace theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.5 Integrals and zeta residues . . . . . . . . . . . . . . . . . . . . . . . 51
6 Quantization and the tangent groupoid 53
6.1 Moyal quantizers and the Moyal deformation . . . . . . . . . . . . . 53
6.2 Smooth groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.3 The tangent groupoid . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.4 Moyal quantization as a continuity condition . . . . . . . . . . . . . . 60
6.5 The hexagon and the analytical index . . . . . . . . . . . . . . . . . . 62
6.6 Quantization and the index theorem . . . . . . . . . . . . . . . . . . 63
7 Equivalence of geometries 65
7.1 Unitary equivalence of spin geometries . . . . . . . . . . . . . . . . . 65
7.2 Morita equivalence and connections . . . . . . . . . . . . . . . . . . 67
7.3 Vector bundles over noncommutative tori . . . . . . . . . . . . . . . 70
7.4 Morita-equivalent toral geometries . . . . . . . . . . . . . . . . . . . 72
7.5 Gauge potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
8 Action functionals 75
8.1 Algebra automorphisms and the metric . . . . . . . . . . . . . . . . . 75
8.2 The fermionic action . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.3 The spectral action principle . . . . . . . . . . . . . . . . . . . . . . 78
8.4 Spectral densities and asymptotics . . . . . . . . . . . . . . . . . . . 79
9 Epilogue: new directions 85
9.1 Noncommutative field theories . . . . . . . . . . . . . . . . . . . . . 85
9.2 Isospectral deformations . . . . . . . . . . . . . . . . . . . . . . . . 86
9.3 Geometries with quantum group symmetry . . . . . . . . . . . . . . 90
9.4 Other developments . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
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