Conformal groups in geometry and spin structures

Conformal groups play a key role in geometry and spin structures. This book provides a self-contained overview of this important area of mathematical physics, beginning with its origins in the works of Cartan and Chevalley and progressing to recent research in spinors and conformal geometry.

Key topics and features:

* Focuses initially on the basics of Clifford algebras

* Studies the spaces of spinors for some even Clifford algebras

* Examines conformal spin geometry, beginning with an elementary study of the conformal group of the Euclidean plane

* Treats covering groups of the conformal group of a regular pseudo-Euclidean space, including a section on the complex conformal group

* Introduces conformal flat geometry and conformal spinoriality groups, followed by a systematic development of riemannian or pseudo-riemannian manifolds having a conformal spin structure

* Discusses links between classical spin structures and conformal spin structures in the context of conformal connections

* Examines pseudo-unitary spin structures and pseudo-unitary conformal spin structures using the Clifford algebra associated with the classical pseudo-unitary space

* Ample exercises with many hints for solutions

* Comprehensive bibliography and index

This text is suitable for a course in mathematical physics at the advanced undergraduate and graduate levels. It will also benefit researchers as a reference text.

Jaime KellerJose Berlin

Foreword	p. ix
Foreword	p. xi
Preface	p. xiii
Overview	p. xix
1 Classic Groups: Clifford Algebras, Projective Quadrics, and Spin Groups	p. 1
1.1 Classical Groups	p. 1
1.1.1 General Linear Groups	p. 1
1.1.2 Symplectic Groups: Classical Results	p. 4
1.1.3 Classical Algebraic Results	p. 4
1.1.4 Classic Groups over Noncommutative Fields	p. 7
1.2 Clifford Algebras	p. 11
1.2.1 Elementary Properties of Quaternion Algebras	p. 11
1.2.2 Clifford Algebras	p. 13
1.3 Involutions of Algebras	p. 23
1.3.1 Classical Definitions	p. 23
1.3.2 T-Symmetric and T-Skew Quantities	p. 23
1.3.3 Involutions over G of a Simple Algebra	p. 24
1.4 Clifford Algebras for Standard Pseudo-Euclidean Spaces E[subscript r,s] and Real Projective Associated Quadrics	p. 26
1.4.1 Clifford Algebras C[subscript r,s] and [Characters not reproducible]: A Review of Standard Definitions	p. 26
1.4.2 Classification of Clifford Algebras C[subscript r,s] and C[Characters not reproducible]	p. 27
1.4.3 Real Projective Quadrics Q(E[subscript r,s])	p. 29
1.5 Pseudoquaternionic Structures on the Space S of Spinors for [Characters not reproducible] m = 2k + 1, r - s [congruent with] [plus or minus] (mod 8). Embedding of Corresponding Spin Groups SpinE[subscript r,s] and Real Projective Quadrics Q(E[subscript r,s])	p. 32
1.5.1 Quaternionic Structures on Right Vector Spaces over H	p. 32
1.5.2 Invariant Scalar Products on Spaces S of Spinors	p. 36
1.5.3 Involutions on the Real Algebra L[subscript H] (S) where S is a Quaternionic Right Vector Space on H, with dim[subscript H]S = n	p. 38
1.5.4 Quaternionic Structures on the Space S of Spinors for [Characters not reproducible] r + s = m = 2k + l, r - s [congruent with] [plus or minus]3 (mod 8)	p. 41
1.5.5 Embedding of Projective Quadrics	p. 44
1.6 Real Structures on the Space S of Spinors for [Characters not reproducible] m = 2k + 1, r - s [congruent with] [plus or minus]1 (mod 8). Embedding of Corresponding Spin Groups and Associated Real Projective Quadrics	p. 48
1.6.1 Involutions of the Real Algebra [pound subscript R] (S) where S is a Real Space over R of Even Dimension	p. 48
1.6.2 Real Symplectic or Pseudo-Euclidean Structures on the Space S of Spinors for [Characters not reproducible] m = r + s = 2k + 1, r - s ][Characters not reproducible] [congruent with] [plus or minus]1 (mod 8)	p. 49
1.6.3 Embedding of Corresponding Projective Quadrics	p. 51
1.7 Study of the Cases r - s [congruent with] 0 (mod 8) and r - s [congruent with] 4 (mod 8)	p. 52
1.7.1 Study of the Case r - s = 0 (mod 8)	p. 52
1.7.2 Study of the Case r - s = 4 (mod 8)	p. 56
1.8 Study of the Case r - s [congruent with] [plus or minus]2 (mod 8)	p. 57
1.8.1 Involutions on A = [pound]c(S), where S is a Complex Vector Space of Dimension n	p. 58
1.8.2 Associated Form with an Involution [alpha] of A = [pound]c(S)	p. 58
1.8.3 Pseudo-Hermitian Structures on the Spaces of Spinors S for [Characters not reproducible] (r - s ][Characters not reproducible] [congruent with] [plus or minus] 2 (mod 8))	p. 58
1.8.4 Embedding of the Corresponding Projective Quadric Q (E[subscript r,s])	p. 59
1.8.5 Concluding Remarks	p. 60
1.9 Appendix	p. 60
1.10 Exercises	p. 61
1.11 Bibliography	p. 68
2 Real Conformal Spin Structures	p. 71
2.1 Some Historical Remarks	p. 71
2.2 Mobius Geometry	p. 74
2.2.1 Mobius Geometry: A Summary of Classical Results	p. 74
2.3 Standard Classical Conformal Plane Geometry	p. 77
2.4 Construction of Covering Groups for the Conformal Group C[subscript n](p, q) of a Standard Pseudo-Euclidean Space E[subscript n](p, q)	p. 78
2.4.1 Conformal Compactification of Standard Pseudo-Euclidean Spaces E[subscript n](p, q)	p. 78
2.4.2 Covering Groups of Conf(E[subscript n] (p, q)) = C[subscript n](p, q)	p. 79
2.4.3 Covering groups of the complex conformal group C[subscript n]	p. 90
2.5 Real Conformal Spinoriality Groups and Flat Real Conformal Geometry	p. 92
2.5.1 Conformal Spinoriality Groups	p. 92
2.5.2 Flat Conformal Spin Structures in Even Dimension	p. 102
2.5.3 Case n = 2r + l, r [greater than] 1	p. 104
2.6 Real Conformal Spin Structures on Manifolds	p. 105
2.6.1 Definitions	p. 105
2.6.2 Manifolds of Even Dimension Admitting a Real Conformal Spin Structure in a Strict Sense	p. 107
2.6.3 Necessary Conditions for the Existence of a Real Conformal Spin Structure in a Strict Sense on Manifolds of Even Dimension	p. 109
2.6.4 Sufficient Conditions for the Existence of Real Conformal Spin Structures in a Strict Sense on Manifolds of Even Dimension	p. 112
2.6.5 Manifolds of Even Dimension with a Real Conformal Spin Structure in a Broad Sense	p. 116
2.6.6 Manifolds of Odd Dimension Admitting a Conformal Spin Special Structure	p. 116
2.7 Links between Spin Structures and Conformal Spin Structures	p. 117
2.7.1 First Links	p. 117
2.7.2 Other Links	p. 118
2.8 Connections: A Review of General Results	p. 119
2.8.1 General Definitions	p. 119
2.8.2 Parallelism	p. 121
2.8.3 Curvature Form and Structure Equation	p. 122
2.8.4 Extensions and Restrictions of Connections	p. 123
2.8.5 Cartan Connections	p. 125
2.8.6 Soudures (Solderings)	p. 125
2.8.7 Ehresmann Connections	p. 126
2.8.8 Ehresmann Connection in a Differentiable Bundle with Structure Group G, a Lie Group	p. 130
2.9 Conformal Ehresmann and Conformal Cartan Connections	p. 132
2.9.1 Conformal Ehresmann Connections	p. 132
2.9.2 Cartan Conformal Connections	p. 138
2.10 Conformal Geodesics	p. 152
2.10.1 Cross Sections and Moving Frames: A Review of Previous Results	p. 152
2.10.2 Conformal Moving Frames	p. 155
2.10.3 The Theory of Yano	p. 158
2.10.4 Conformal Normal Frames Associated with a Curve	p. 159
2.10.5 Conformal Geodesies	p. 160
2.11 Generalized Conformal Connections	p. 163
2.11.1 Conformal Development	p. 163
2.11.2 Generalized Conformal Connections	p. 172
2.12 Vahlen Matrices	p. 181
2.12.1 Historical Background	p. 181
2.12.2 Study of Classical Mobius Transformations of R[superscript n]	p. 182
2.12.3 Study of the Anti-Euclidean Case E[subscript n]-1 (0, n - 1)	p. 183
2.12.4 Study of Indefinite Quadratic Spaces	p. 184
2.13 Exercises	p. 186
2.14 Bibliography	p. 199
3 Pseudounitary Conformal Spin Structures	p. 205
3.1 Pseudounitary Conformal Structures	p. 206
3.1.1 Introduction	p. 206
3.1.2 Algebraic Characterization	p. 206
3.1.3 Some remarks about the Standard Group U(p, q)	p. 208
3.1.4 An Algebraic Recall	p. 208
3.1.5 Connectedness	p. 208
3.1.6 General Definitions	p. 208
3.2 Projective Quadric Associated with a Pseudo-Hermitian Standard Space H[subscript p,q]	p. 209
3.3 Conformal Compactification of Pseudo-Hermitian Standard Spaces H[subscript p,q], p + q = n	p. 210
3.3.1 Introduction	p. 210
3.4 Pseudounitary Conformal Groups of Pseudo-Hermitian Standard Spaces H[subscript p,q]	p. 212
3.4.2 Translations of E	p. 213
3.4.3 Dilatations of E and the Pseudounitary Group Sim U(p, q)	p. 213
3.4.4 Algebraic Characterization	p. 214
3.5 The Real Conformal Symplectic Group and the Pseudounitary Conformal Group	p. 216
3.5.1 Definition of the Real Conformal Symplectic Group	p. 216
3.6 Topology of the Projective Quadrics H[subscript p,q]	p. 217
3.6.1 Topological Properties	p. 217
3.6.2 Generators of the Projective Quadrics H[subscript p,q]	p. 219
3.7 Clifford Algebras and Clifford Groups of Standard Pseudo-Hermitian Spaces H[subscript p,q]	p. 219
3.7.1 Fundamental Algebraic Properties	p. 219
3.7.2 Definition of the Clifford Algebra Associated with H [subscript p,q]	p. 221
3.7.3 Definition 2 of the Clifford Algebra Associated with H [subscript p, q]	p. 224
3.7.4 Clifford Groups and Covering Groups of U (p, q)	p. 226
3.7.5 Fundamental Diagram Associated with RU (p, q)	p. 227
3.7.6 Characterization of U(p, q)	p. 229
3.7.7 Associated Spinors	p. 231
3.8 Natural Embeddings of the Projective Quadrics H[subscript p,q]	p. 233
3.9 Covering Groups of the Conformal Pseudounitary Group	p. 234
3.9.1 A Review of Previous Results	p. 234
3.9.2 Algebraic Construction of Covering Groups PU(F)	p. 234
3.9.3 Conformal Flat Geometry (n = p + q = 2r)	p. 236
3.9.4 Pseudounitary Flat Spin Structures and Pseudounitary Conformal Flat Spin Structures	p. 240
3.9.5 Study of the Case n = p + q = 2r + 1	p. 242
3.10 Pseudounitary Spinoriality Groups and Pseudounitary Conformal Spinoriality Groups	p. 242
3.10.1 Classical Spinoriality Groups	p. 242
3.10.2 Pseudounitary Spinoriality Groups	p. 243
3.10.3 Pseudounitary Conformal Spinoriality Groups	p. 244
3.11 Pseudounitary Spin Structures on a Complex Vector Bundle	p. 245
3.11.1 Review of Complex Pseudo-Hermitian Vector Bundles	p. 245
3.11.2 Pseudounitary Spin Structures on a Complex Vector Bundle	p. 245
3.11.3 Obstructions to the Existence of Spin Structures	p. 246
3.11.4 Definition of the Fundamental Pseudounitary Bundle	p. 246
3.12 Pseudonitary Spin Structures and Pseudounitary Conformal Spin Structures on an Almost Complex 2n-Dimensional Manifold V	p. 250
3.12.1 Pseudounitary Spin Structures	p. 250
3.12.2 Necessary Conditions for the Existence of a Pseudonitary Spin Structure in a Strict Sense on V	p. 252
3.12.3 Sufficient Conditions for the Existence of a Pseudounitary Spin Structure in a Strict Sense on V	p. 252
3.12.4 Manifolds V With a Pseudounitary Spin Structure in a Broad Sense	p. 253
3.12.5 Pseudounitary Conformal Spin Structures	p. 253
3.12.6 Links between Pseudounitary Spin Structures and Pseudounitary Conformal Spin Structures	p. 256
3.12.7 Concluding Remarks	p. 257
3.13 Appendix	p. 257
3.13.1 A Review of Algebraic Topology	p. 257
3.13.2 Complex Operators and Complex Structures Pseudo-Adapted to a Symplectic Form	p. 258
3.13.3 Some Comments about Spinoriality Groups	p. 261
3.14 Exercises	p. 263
3.15 Bibliography	p. 269
Index	p. 275

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