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Esposito G. — Dirac Operators and Spectral Geometry
Esposito G. — Dirac Operators and Spectral Geometry

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Название: Dirac Operators and Spectral Geometry

Автор: Esposito G.

Аннотация:

The Dirac operator has many useful applications in theoretical physics and mathematics. This book provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local or nonlocal boundary conditions. The theory is introduced at a level suitable for graduate students. Numerous examples are then given to illustrate the peculiar properties of the Dirac operator, and the role of boundary conditions in heat-kernel asymptotics and quantum field theory. Topics covered include the introduction of spin-structures in Riemannian and Lorentzian manifolds; applications of index theory; heat-kernel asymptotics for operators of Laplace type; quark boundary conditions; one-loop quantum cosmology; conformally covariant operators; and the role of the Dirac operator in some recent investigations of four-manifolds. This volume provides graduate students with a rigorous introduction and researchers with a invaluable reference to the Dirac operator and its applications in theoretical physics.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Cambridge University Press

Год издания: 1998

Количество страниц: 219

Добавлена в каталог: 07.02.2011

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Adjoint action      57
Anti-commuting complex structures on $\mathfrak{R}^{n}$      62
Anti-self-dual gauge curvature      189
Bag model      150
Ball-bundle      42
Bessel functions of large order (uniform asymptotics)      101—105
Bott periodicity theorem for the infinite orthogonal group      63
Bott periodicity theorem for the unitary group      60
Bott theorem (global version)      64 67
Bott theorem (local form)      67
Boundary conditions in the Eastwood — Singer gauge      157—159
Boundary invariants      114 115
Boundary operator      110
Boundary operator in Euclidean quantum gravity      168
Bulk states      40
Characteristic polynomila      71
Chern character      72
Chern numbers      71
Chiral representation      6
Clifford algebra      24
Clifford group      25
Collar      68
Complex conjugation of spinors      95
Conformal anomaly for a massless spin-$\frac{1}{2}$ field      89 153
Conformal Laplacian      163
Conformal-variation formulae      117 186
Conformally covariant operators      163
Continuous functor      116
Covariant derivative of spinor fields      7
Cup-product      27
Degree of a family of Dirac operators      53
Dirac equation      2 4 5
Dirac genus      51
Dirac representation      5
Direct limit with the fine topology      63
Dirichlet boundary conditions      107 109 113 137
Eastwood — Singer gauge in curved space      161
Eastwood — Singer gauge in flat space      156
Electric boundary conditions for Euclidean Maxwell theory      155
Elliptic operator      26
Ellipticity of a system of boundary operators      44
Ellipticity of a system of differential operators      44
Equivalent symbols      64
Eta function      76 82
Euclidean conjugation      100
Euler number      75
Exact sequence      70
Extended Dirac operator      47
Extrinsic-curvature tensor      113
Fourth-order ghost operator      159
Fredholm operators      32
Functional determinant of a conformally covariant operator      164
Gamma function      81
Gamma matrices      5 207
Gauge-field operator in the Eastwood — Singer gauge      157
Gauge-invariant boundary conditions in one-loop quantum gravity      166 167
Gauss — Bonnet — Chern theorem      75
General structure of the $\zeta (0)$ value      88
Genus      72
Ghost basis functions in the Eastwood — Singer gauge      160 161
Ghost fields      185
Graded leading symbol      69
Graded vector bundle      68
Gravitational index formula      46
Green's kernel      110
Group of gauge transformations      189
Harmonic forms      24
Harmonic spinors      29
Heat kernel      111
Heat-kernel diagonal      112
Hirzebruch L-polynomial      72
Homomorphism of groups      19
Index formula for gauge theories      47 51
Index of elliptic operators      26
Index of families of elliptic operators      52
Index theorem and $\zeta$-functions      131
Index theorem and heat kernels      129
Index theorem for a closed manifold      43
Index theorem for global boundary conditions      76
Infeld — van der Waerden symbols      96
Infinite orthogonal group      63
Inner automorphism of the unitary group      56
Integrated heat kernel      110
Interior invariants      114 115
Intersection form      189
Intrinsic metric on the boundary      112
Invariant for the Paneitz operator      164
Inverse eigenvalue problem for vibrating membranes      107—109 137
Inverse Mellin transform      130
Involutions on Clifford algebras      24
Kernel with non-local boundary conditions      80
Lagrangian for a Dirac field      8
Lagrangian of the supersymmetric non-linear $\sigma$-model      48
Laplace operator      24
Laplace operator on scalar functions      162
Laplace-type operator      22 23 111
Leading symbol      26
Left-invariant Riemannian metric on U(n)      56
Levi sum      135
Lichnerowicz — Weitzenboeck formula      175
Lie algebra of SU(n)      57
Limit circle at zero      91
Limit point case at zero      91
Local boundary conditions of Breitenlohner, Freedman and Hawking      152
Local boundary conditions of Luckock et al.      152
Local boundary conditions proposed for quark confinement      151
Local flux of colour and flavour quantum numbers      151
Magnetic boundary conditions for Euclidean Maxwell theory      155
Majorana representation      5
Mixed boundary conditions      126
Moduli space      189
Motivations for studying manifolds with boundary      149 150
Motivations for studying quantum gravity      165 166
Non-local functionals in the path integral      184
Out-in amplitude      184
Paneitz operator      163
Pauli matrices      17
Periodic boundary conditions      30 31 82
Pontrjagin classes      72
Pure tones      137
Quantum theory of fermionic fields      84
Quark-gluon fields      150
Riemann tensor      6
Riemannian volume element      110 132
Robin boundary conditions      113
Rotation      16
Schroedinger equation      2 143
Schroedinger kernel      144
Seiberg — Witten equations      176 178
Seiberg — Witten invariants      178 179
Self-dual gauge curvature      188
Semi-group      65
Short exact sequence      70
Signature operator      27
Soldering form      97
Space-time      6
Spectral boundary conditions      85 86
Spectral flow      30
Spectral resolution      119
Sphere-bundle      42
Spin-connection      7
Spin-manifold      11
Spin-representation      14
Spin-space      95
Spin-structure (for Riemannian manifolds)      9 10
Spinor covariant derivative      99
Spinor fields      7 11
Spinor structure in Lorentzian four-manifolds      9
Split of the tangent bundle      68
Stably equivalent bundles      66
Stiefel — Whitney classes      73
Strong ellipticity      69
Supersymmetry algebra      47
Surface states      40
Theta function      32
Todd class      72
Topology of the moduli space      190
Total Chern form      71
Total Dirac operator      14 28
Triadic von Koch snowflake      145 146
Universal constants      113
Universal functions in heat-kernel asymptotics      170
Varadhan estimate      133
Vierbein      6
Weyl calculus      36
Weyl tensor      163
Weyl tensor (spinor form)      98
Weyl's formula      138 141
Weyl's limit point — limit circle criterion      91
Yang — Mills equations      188
Yang — Mills functional      188
Zeta function      130
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