Dirac Operators and Spectral Geometry

Contents......Page ix
Preface......Page xi
Acknowledgments......Page xiii
1.1 The Dirac equation......Page p1
1.2 Spinor fields in Lorentzian manifolds......Page p6
1.3 Spin-structures: Riemannian case......Page p9
1.4 Global theory of the Dirac operator......Page p12
Appendix 1.A......Page p15
2.1 Operators of Laplace type......Page p22
2.2 Clifford algebras......Page p24
2.3 Index of elliptic operators and signature operator......Page p25
2.4 Dirac operator......Page p28
2.5 Some outstanding problems......Page p33
2.6 Pseudo-differential operators......Page p35
3.1 Index problem for manifolds with boundary......Page p41
3.2 Elliptic boundary conditions......Page p44
3.3 Index theorems and gauge fields......Page p45
3.4 Index of two-parameter families......Page p52
3.5 Determinants of Dirac operators......Page p53
3.6 Bott periodicity......Page p55
3.7 K-theory......Page p65
Appendix 3.A......Page p68
Appendix 3.B......Page p70
4 Spectral asymmetry......Page p74
4.1 Spectral asymmetry and Riemannian geometry......Page p75
4.2 Eta(0) calculation......Page p77
4.3 A further example......Page p82
4.4 Massless spin-1/2 fields......Page p84
4.5 Zeta(0) value with non-local boundary conditions......Page p86
4.6 Essential self-adjointness......Page p90
Appendix 4.A......Page p95
Appendix 4.B......Page p101
5 Spectral geometry with operators of Laplace type......Page p106
5.1 On hearing the shape of a drum......Page p107
5.2 The Laplacian on manifolds with boundary......Page p111
5.3 Functorial method......Page p115
5.4 Mixed boundary conditions......Page p126
5.5 Heat equation and index theorem......Page p128
5.6 Heat kernel and zeta-function......Page p130
5.7 Majorizations of the heat kernel......Page p131
Appendix 5.A......Page p136
6.1 Introduction......Page p148
6.2 Quark boundary conditions......Page p150
6.3 Quantum cosmology......Page p152
6.4 Conformal gauges and manifolds with boundary......Page p154
6.5 Conformally covariant operators......Page p162
6.6 Euclidean quantum gravity......Page p165
6.7 Dirac operator and the theory of four-manifolds......Page p174
Appendix 6.A......Page p181
Appendix 6.B......Page p183
Appendix 6.C......Page p186
Appendix 6.D......Page p187
References......Page p191
Index......Page p207