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Index
Preface
Acknowledgements
Introduction
Part I Relativistic field theory in Minkowski spacetime
1 Scalar field theory and its symmetries
1.1 The scalar field system
1.2 Symmetries of the system
1.2.1 SO( n ) internal symmetry
1.2.2 General internal symmetry
1.2.3 Spacetime symmetries – the Lorentz and Poincaré groups
1.3 Noether currents and charges
1.4 Symmetries in the canonical formalism
1.5 Quantum operators
1.6 The Lorentz group for D = 4
2 The Dirac field
2.1 The homomorphism of SL(2,
) → SO(3, 1)
2.2 The Dirac equation
2.3 Dirac adjoint and bilinear form
2.4 Dirac action
2.5 The spinors u (
, s ) and υ (
, s ) for D = 4
2.6 Weyl spinor fields in even spacetime dimension
2.7 Conserved currents
2.7.1 Conserved U(1) current
2.7.2 Energy–momentum tensors for the Dirac field
3 Clifford algebras and spinors
3.1 The Clifford algebra in general dimension
3.1.1 The generating
-matrices
3.1.2 The complete Clifford algebra
3.1.3 Levi-Civita symbol
3.1.4 Practical
-matrix manipulation
3.1.5 Basis of the algebra for even dimension D = 2 m
3.1.6 The highest rank Clifford algebra element
3.1.7 Odd spacetime dimension D = 2 m + 1
3.1.8 Symmetries of
-matrices
3.2 Spinors in general dimensions
3.2.1 Spinors and spinor bilinears
3.2.2 Spinor indices
3.2.3 Fierz rearrangement
3.2.4 Reality
3.3 Majorana spinors
3.3.1 Definition and properties
3.3.2 Symplectic Majorana spinors
3.3.3 Dimensions of minimal spinors
3.4 Majorana spinors in physical theories
3.4.1 Variation of a Majorana Lagrangian
3.4.2 Relation of Majorana and Weyl spinor theories
3.4.3 U(1) symmetries of a Majorana field
Appendix 3A Details of the Clifford algebras for D = 2 m
3A.1 Traces and the basis of the Clifford algebra
3A.2 Uniqueness of the
-matrix representation
3A.3 The Clifford algebra for odd spacetime dimensions
3A.4 Determination of symmetries of
-matrices
3A.5 Friendly representations
4 The Maxwell an Yang–Mills gauge fields
4.1 The abelian gauge field
4.1.1 Gauge invariance and fields with electric charge
4.1.2 The free gauge field
4.1.3 Sources and Green’s function
4.1.4 Quantum electrodynamics
4.1.5 The stress tensor and gauge covariant translations
4.2 Electromagnetic duality
4.2.1 Dual tensors
4.2.2 Duality for one free electromagnetic field
4.2.3 Duality for gauge field and complex scalar
4.2.4 Electromagnetic duality for coupled Maxwell fields
4.3 Non-abelian gauge symmetry
4.3.1 Global internal symmetry
4.3.2 Gauging the symmetry
4.3.3 Yang–Mills field strength and action
4.3.4 Yang–Mills theory for G = SU( N )
4.4 Internal symmetry for Majorana spinors
5 The free Rarita–Schwinger field
5.1 The initial value problem
5.2 Sources and Green’s function
5.3 Massive gravitinos from dimensional reduction
5.3.1 Dimensional reduction for scalar fields
5.3.2 Dimensional reduction for spinor fields
5.3.3 Dimensional reduction for the vector gauge field
5.3.4 Finally
6
= 1 global supersymmetry in D = 4
6.1 Basic SUSY field theory
6.1.1 Conserved supercurrents
6.1.2 SUSY Yang–Mills theory
6.1.3 SUSY transformation rules
6.2 SUSY field theories of the chiral multiplet
6.2.1 U(1) R symmetry
R
6.2.2 The SUSY algebra
6.2.3 More chiral multiplets
6.3 SUSY gauge theories
6.3.1 SUSY Yang–Mills vector multiplet
6.3.2 Chiral multiplets in SUSY gauge theories
6.4 Massless representations of
-extended supersymmetry
6.4.1 Particle representations of
-extended supersymmetry
6.4.2 Structure of massless representations
Appendix 6A Extended supersymmetry and Weyl spinors
Appendix 6B On- and off-shell multiplets and degrees of freedom
Part II Differential geometry and gravity
7 Differential geometry
7.1 Manifolds
7.2 Scalars, vectors, tensors, etc.
7.3 The algebra and calculus of differential forms
7.4 The metric and frame field on a manifold
7.4.1 The metric
7.4.2 The frame field
7.4.3 Induced metrics
7.5 Volume forms and integration
7.6 Hodge duality of forms
7.7 Stokes’ theorem and electromagnetic charges
7.8 p -form gauge fields
7.9 Connections and covariant derivatives
7.9.1 The first structure equation and the spin connection
7.9.2 The affine connection
7.9.3 Partial integration
7.10 The second structure equation and the curvature tensor
7.11 The nonlinear
-model
7.12 Symmetries and Killing vectors
7.12.1
-model symmetries
7.12.2 Symmetries of the Poincaré plane
8 The first and second order formulations of general relativity
8.1 Second order formalism for gravity and bosonic matter
8.2 Gravitational fluctuations of flat spacetime
8.2.1 The graviton Green’s function
8.3 Second order formalism for gravity and fermions
8.4 First order formalism for gravity and fermions
Part III Basic supergravity
9
= 1 pure supergravity in four dimensions
9.1 The universal part of supergravity
9.2 Supergravity in the first order formalism
9.3 The 1.5 order formalism
9.4 Local supersymmetry of
= 1, D = 4 supergravity
9.5 The algebra of local supersymmetry
9.6 Anti-de Sitter supergravity
10 D = 11 supergravity
10.1 D ≤ 11 from dimensional reduction
10.2 The field content of D = 11 supergravity
10.3 Construction of the action and transformation rules
10.4 The algebra of D = 11 supergravity
11 General gauge theory
11.1 Symmetries
11.1.1 Global symmetries
11.1.2 Local symmetries and gauge fields
11.1.3 Modified symmetry algebras
11.2 Covariant quantities
11.2.1 Covariant derivatives
11.2.2 Curvatures
11.3 Gauged spacetime translations
11.3.1 Gauge transformations for the Poincaré group
11.3.2 Covariant derivatives and general coordinate transformations
11.3.3 Covariant derivatives and curvatures in a gravity theory
11.3.4 Calculating transformations of covariant quantities
Appendix 11A Manipulating covariant derivatives
11A.1 Proof of the main lemma
11A.2 Examples in supergravity
12 Survey of supergravities
12.1 The minimal superalgebras
12.1.1 Four dimensions
12.1.2 Minimal superalgebras in higher dimensions
12.2 The R -symmetry group
12.3 Multiplets
12.3.1 Multiplets in four dimensions
12.3.2 Multiplets in more than four dimensions
12.4 Supergravity theories: towards a catalogue
12.4.1 The basic theories and kinetic terms
12.4.2 Deformations and gauged supergravities
12.5 Scalars and geometry
12.6 Solutions and preserved supersymmetries
12.6.1 Anti-de Sitter superalgebras
12.6.2 Central charges in four dimensions
12.6.3 ‘Central charges’ in higher dimensions
Part IV Complex geometry and global SUSY
13 Complex manifolds
13.1 The local description of complex and Kähler manifolds
13.2 Mathematical structure of Kähler manifolds
13.3 The Kähler manifolds CP n
n
13.4 Symmetries of Kähler metrics
13.4.1 Holomorphic Killing vectors and moment maps
13.4.2 Algebra of holomorphic Killing vectors
13.4.3 The Killing vectors of CP 1
1
14 General actions with
= 1 supersymmetry
14.1 Multiplets
14.1.1 Chiral multiplets
14.1.2 Real multiplets
14.2 Generalized actions by multiplet calculus
14.2.1 The superpotential
14.2.2 Kinetic terms for chiral multiplets
14.2.3 Kinetic terms for gauge multiplets
14.3 Kähler geometry from chiral multiplets
14.4 General couplings of chiral multiplets and gauge multiplets
14.4.1 Global symmetries of the SUSY
-model
14.4.2 Gauge and SUSY transformations for chiral multiplets
14.4.3 Actions of chiral multiplets in a gauge theory
14.4.4 General kinetic action of the gauge multiplet
14.4.5 Requirements for an
= 1 SUSY gauge theory
14.5 The physical theory
14.5.1 Elimination of auxiliary fields
14.5.2 The scalar potential
14.5.3 The vacuum state and SUSY breaking
14.5.4 Supersymmetry breaking and the Goldstone fermion
14.5.5 Mass spectra and the supertrace sum rule
14.5.6 Coda
Appendix 14A Superspace
Appendix 14B Appendix: Covariant supersymmetry transformations
Part V Superconformal construction of supergravity theories
15 Gravity as a conformal gauge theory
15.1 The strategy
15.2 The conformal algebra
15.3 Conformal transformations on fields
15.4 The gauge fields and constraints
15.5 The action
15.6 Recapitulation
15.7 Homothetic Killing vectors
16 The conformal approach to pure
= 1 supergravity
16.1 Ingredients
16.1.1 Superconformal algebra
16.1.2 Gauge fields, transformations, and curvatures
16.1.3 Constraints
16.1.4 Superconformal transformation rules of a chiral multiplet
16.2 The action
16.2.1 Superconformal action of the chiral multiplet
16.2.2 Gauge fixing
16.2.3 The result
17 Construction of the matter-coupled
= 1 supergravity
17.1 Superconformal tensor calculus
17.1.1 The superconformal gauge multiplet
17.1.2 The superconformal real multiplet
17.1.3 Gauge transformations of superconformal chiral multiplets
17.1.4 Invariant actions
17.2 Construction of the action
17.2.1 Conformal weights
17.2.2 Superconformal invariant action (ungauged)
17.2.3 Gauged superconformal supergravity
17.2.4 Elimination of auxiliary fields
17.2.5 Partial gauge fixing
17.3 Projective Kähler manifolds
17.3.1 The example of CP n
n
17.3.2 Dilatations and holomorphic homothetic Killing vectors
17.3.3 The projective parametrization
17.3.4 The Kähler cone
17.3.5 The projection
17.3.6 Kähler transformations
17.3.7 Physical fermions
17.3.8 Symmetries of projective Kähler manifolds
17.3.9 T -gauge and decomposition laws
17.3.10 An explicit example: SU(1, 1)/U(1) model
17.4 From conformal to Poincaré supergravity
17.4.1 The superpotential
17.4.2 The potential
17.4.3 Fermion terms
17.5 Review and preview
17.5.1 Projective and Kähler–Hodge manifolds
17.5.2 Compact manifolds
Appendix 17A Kähler–Hodge manifolds
17A.1 Dirac quantization condition
17A.2 Kähler–Hodge manifolds
Appendix 17B Steps in the derivation of (17.7)
Part VI
= 1 supergravity actions and applications
18 The physical
= 1 matter-coupled supergravity
18.1 The physical action
18.2 Transformation rules
18.3 Further remarks
18.3.1 Engineering dimensions
18.3.2 Rigid or global limit
18.3.3 Quantum effects and global symmetries
19 Applications of
= 1 supergravity
19.1 Supersymmetry breaking and the super-BEH effect
19.1.1 Goldstino and the super-BEH effect
19.1.2 Extension to cosmological solutions
19.1.3 Mass sum rules in supergravity
19.2 The gravity mediation scenario
19.2.1 The Polónyi model of the hidden sector
19.2.2 Soft SUSY breaking in the observable sector
19.3 No-scale models
19.4 Supersymmetry and anti-de Sitter space
19.5 R -symmetry and Fayet–Iliopoulos terms
19.5.1 The R -gauge field and transformations
19.5.2 Fayet–Iliopoulos terms
19.5.3 An example with non-minimal Kähler potential
Part VII Extended
= 2 supergravity
20 Construction of the matter-coupled
= 2 supergravity
20.1 Global supersymmetry
20.1.1 Gauge multiplets for D = 6
20.1.2 Gauge multiplets for D = 5
20.1.3 Gauge multiplets for D = 4
20.1.4 Hypermultiplets
20.1.5 Gauged hypermultiplets
20.2
= 2 superconformal calculus
20.2.1 The superconformal algebra
20.2.2 Gauging of the superconformal algebra
20.2.3 Conformal matter multiplets
20.2.4 Superconformal actions
20.2.5 Partial gauge fixing
20.2.6 Elimination of auxiliary fields
20.2.7 Complete action
20.2.8 D = 5 and D = 6,
= 2 supergravities
20.3 Special geometry
20.3.1 The family of special manifolds
20.3.2 Very special real geometry
20.3.3 Special Kähler geometry
20.3.4 Hyper-Kähler and quaternionic-Kähler manifolds
20.4 From conformal to Poincaré supergravity
20.4.1 Kinetic terms of the bosons
20.4.2 Identities of special Kähler geometry
20.4.3 The potential
20.4.4 Physical fermions and other terms
20.4.5 Supersymmetry and gauge transformations
Appendix 20A SU(2) conventions and triplets
Appendix 20B Dimensional reduction 6 → 5 → 4
20B.1 Reducing from D = 6 → D = 5
20B.2 Reducing from D = 5 → D = 4
Appendix 20C Definition of rigid special Kähler geometry
21 The physical
= 2 matter-coupled supergravity
21.1 The bosonic sector
21.1.1 The basic (ungauged)
= 2, D = 4 matter-coupled supergravity
21.1.2 The gauged supergravities
21.2 The symplectic formulation
21.2.1 Symplectic definition
21.2.2 Comparison of symplectic and prepotential formulation
21.2.3 Gauge transformations and symplectic vectors
21.2.4 Physical fermions and duality
21.3 Action and transformation laws
21.3.1 Final action
21.3.2 Supersymmetry transformations
21.4 Applications
21.4.1 Partial supersymmetry breaking
21.4.2 Field strengths and central charges
21.4.3 Moduli spaces of Calabi–Yau manifolds
21.5 Remarks
21.5.1 Fayet–Iliopoulos terms
21.5.2
-model symmetries
21.5.3 Engineering dimensions
Part VIII Classical solutions and the AdS/CFT correspondence
22 Classical solutions of gravity and supergravity
22.1 Some solutions of the field equations
22.1.1 Prelude: frames and connections on spheres
22.1.2 Anti-de Sitter space
22.1.3 AdS D obtained from its embedding in
D
D +1
22.1.4 Space time metrics with spherical symmetry
22.1.5 AdS–Schwarzschild spacetime
22.1.6 The Reissner–Nordström metric
22.1.7 A more general Reissner–Nordström solution
22.2 Killing spinors and BPS solutions
22.2.1 The integrability condition for Killing spinors
22.2.2 Commuting and anti-commuting Killing spinors
22.3 Killing spinors for anti-de Sitter space
22.4 Extremal Reissner–Nordström spacetimes as BPS solutions
22.5 The black hole attractor mechanism
22.5.1 Example of a black hole attractor
22.5.2 The attractor mechanism – real slow and simple
22.6 Supersymmetry of the black holes
22.6.1 Killing spinors
22.6.2 The central charge
22.6.3 The black hole potential
22.7 First order gradient flow equations
22.8 The attractor mechanism – fast and furious
Appendix 22A Killing spinors for pp-waves
23 The AdS/CFT correspondence
23.1 The
= 4 SYM theory
23.2 Type IIB string theory and D 3-branes
23.3 The D 3-brane solution of Type IIB supergravity
23.4 Kaluza–Klein analysis on AdS 5
5
S 5
5
23.5 Euclidean AdS and its inversion symmetry
23.6 Inversion and CFT correlation functions
23.7 The free massive scalar field in Euclidean AdS d +1
d +1
23.8 AdS/CFT correlators in a toy model
23.9 Three-point correlation functions
23.10 Two-point correlation functions
23.11 Holographic renormalization
23.11.1 The scalar two-point function in a CFT d
d
23.11.2 The holographic trace anomaly
23.12 Holographic RG flows
23.12.1 AAdS domain wall solutions
23.12.2 The holographic c -theorem
23.12.3 First order flow equations
23.13 AdS/CFT and hydrodynamics
Appendix A Comparison of notation
A.1 Spacetime and gravity
A.2 Spinor conventions
A.3 Components of differential forms
A.4 Covariant derivatives
Appendix B Lie algebras and superalgebras
B.1 Groups and representations
B.2 Lie algebras
B.3 Superalgebras
References
Index
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