You are previewing Supergravity.
O'Reilly logo
Supergravity

Book Description

Supergravity, together with string theory, is one of the most significant developments in theoretical physics. Written by two of the most respected workers in the field, this is the first-ever authoritative and systematic account of supergravity. The book starts by reviewing aspects of relativistic field theory in Minkowski spacetime. After introducing the relevant ingredients of differential geometry and gravity, some basic supergravity theories (D=4 and D=11) and the main gauge theory tools are explained. In the second half of the book, complex geometry and N=1 and N=2 supergravity theories are covered. Classical solutions and a chapter on AdS/CFT complete the book. Numerous exercises and examples make it ideal for Ph.D. students, and with applications to model building, cosmology and solutions of supergravity theories, it is also invaluable to researchers. A website hosted by the authors, featuring solutions to some exercises and additional reading material, can be found at www.cambridge.org/supergravity.

Table of Contents

  1. Cover
  2. Title
  3. Copyright Page
  4. Contents
  5. Preface
  6. Acknowledgements
  7. Introduction
  8. Part I: Relativistic Field Theory in Minkowski Spacetime
    1. 1 - Scalar Field Theory and its Symmetries
      1. 1.1 The Scalar Field System
      2. 1.2 Symmetries of the System
        1. 1.2.1 SO(n) Internal Symmetry
        2. 1.2.2 General Internal Symmetry
        3. 1.2.3 Spacetime Symmetries – The Lorentz and Poincaré Groups
      3. 1.3 Noether Currents and Charges
      4. 1.4 Symmetries in the Canonical Formalism
      5. 1.5 Quantum Operators
      6. 1.6 The Lorentz Group for D = 4
    2. 2 - The Dirac Field
      1. 2.1 The Homomorphism of SL(2, ℂ) → SO(3, 1)
      2. 2.2 The Dirac Equation
      3. 2.3 Dirac Adjoint and Bilinear Form
      4. 2.4 Dirac Action
      5. 2.5 The Spinors u(p, s) and υ(p, s) for D = 4
      6. 2.6 Weyl Spinor Fields in even Spacetime Dimension
      7. 2.7 Conserved Currents
        1. 2.7.1 Conserved U(1) Current
        2. 2.7.2 Energy–Momentum Tensors for the Dirac Field
    3. 3 - Clifford Algebras and Spinors
      1. 3.1 The Clifford Algebra in General Dimension
        1. 3.1.1 The Generating γ -matrices
        2. 3.1.2 The Complete Clifford Algebra
        3. 3.1.3 Levi-Civita Symbol
        4. 3.1.4 Practical γ -matrix Manipulation
        5. 3.1.5 Basis of the Algebra for even Dimension D = 2m
        6. 3.1.6 The Highest Rank Clifford Algebra Element
        7. 3.1.7 Odd Spacetime Dimension D = 2m + 1
        8. 3.1.8 Symmetries of γ -matrices
      2. 3.2 Spinors in General Dimensions
        1. 3.2.1 Spinors and Spinor Bilinears
        2. 3.2.2 Spinor Indices
        3. 3.2.3 Fierz Rearrangement
        4. 3.2.4 Reality
      3. 3.3 Majorana Spinors
        1. 3.3.1 Definition and Properties
        2. 3.3.2 Symplectic Majorana Spinors
        3. 3.3.3 Dimensions of Minimal Spinors
      4. 3.4 Majorana Spinors in Physical Theories
        1. 3.4.1 Variation of a Majorana Lagrangian
        2. 3.4.2 Relation of Majorana and Weyl Spinor Theories
        3. 3.4.3 U(1) Symmetries of a Majorana Field
      5. Appendix 3A Details of the Clifford Algebras for D = 2m
        1. 3A.1 Traces and the Basis of the Clifford Algebra
        2. 3A.2 Uniqueness of the γ -matrix Representation
        3. 3A.3 The Clifford Algebra for Odd Spacetime Dimensions
        4. 3A.4 Determination of Symmetries of γ -matrices
        5. 3A.5 Friendly Representations
    4. 4 - The Maxwell an Yang–Mills gauge fields
      1. 4.1 The Abelian Gauge Field Aμ(x)
        1. 4.1.1 Gauge Invariance and Fields with Electric Charge
        2. 4.1.2 The Free Gauge Field
        3. 4.1.3 Sources and Green’s Function
        4. 4.1.4 Quantum Electrodynamics
        5. 4.1.5 The Stress Tensor and Gauge Covariant Translations
      2. 4.2 Electromagnetic Duality
        1. 4.2.1 Dual Tensors
        2. 4.2.2 Duality for one Free Electromagnetic Field
        3. 4.2.3 Duality for Gauge Field and Complex Scalar
        4. 4.2.4 Electromagnetic Duality for Coupled Maxwell Fields
      3. 4.3 Non-Abelian Gauge Symmetry
        1. 4.3.1 Global Internal Symmetry
        2. 4.3.2 Gauging the Symmetry
        3. 4.3.3 Yang–Mills Field Strength and Action
        4. 4.3.4 Yang–Mills Theory for G = SU(N)
      4. 4.4 Internal Symmetry for Majorana Spinors
    5. 5 - The Free Rarita–Schwinger Field
      1. 5.1 The Initial Value Problem
      2. 5.2 Sources and Green’s Function
      3. 5.3 Massive Gravitinos from Dimensional Reduction
        1. 5.3.1 Dimensional Reduction for Scalar Fields
        2. 5.3.2 Dimensional Reduction for Spinor Fields
        3. 5.3.3 Dimensional Reduction for the Vector Gauge Field
        4. 5.3.4 Finally Ψμ(x, y)
    6. 6 - N = 1 Global Supersymmetry in D = 4
      1. 6.1 Basic SUSY Field Theory
        1. 6.1.1 Conserved Supercurrents
        2. 6.1.2 SUSY Yang–Mills Theory
        3. 6.1.3 SUSY Transformation Rules
      2. 6.2 SUSY Field Theories of the Chiral Multiplet
        1. 6.2.1 U(1)R Symmetry
        2. 6.2.2 The SUSY Algebra
        3. 6.2.3 More Chiral Multiplets
      3. 6.3 SUSY Gauge Theories
        1. 6.3.1 SUSY Yang–Mills Vector Multiplet
        2. 6.3.2 Chiral Multiplets in SUSY Gauge Theories
      4. 6.4 Massless Representations of N-extended Supersymmetry
        1. 6.4.1 Particle Representations of N-extended Supersymmetry
        2. 6.4.2 Structure of Massless Representations
      5. Appendix 6A Extended Supersymmetry and Weyl Spinors
        1. Appendix 6B On- and Off-Shell Multiplets and Degrees of Freedom
  9. Part II: Differential Geometry and Gravity
    1. 7 - Differential Geometry
      1. 7.1 Manifolds
      2. 7.2 Scalars, Vectors, Tensors, etc.
      3. 7.3 The Algebra and Calculus of Differential Forms
      4. 7.4 The Metric and Frame Field on a Manifold
        1. 7.4.1 The Metric
        2. 7.4.2 The Frame Field
        3. 7.4.3 Induced Metrics
      5. 7.5 Volume Forms and Integration
      6. 7.6 Hodge Duality of Forms
      7. 7.7 Stokes’ Theorem and Electromagnetic Charges
      8. 7.8 p-Form gauge fields
      9. 7.9 Connections and Covariant Derivatives
        1. 7.9.1 The First Structure Equation and the Spin Connection ωμab
        2. 7.9.2 The Affine Connection Γρμv
        3. 7.9.3 Partial Integration
      10. 7.10 The Second Structure Equation and the Curvature Tensor
      11. 7.11 The Nonlinear σ-model
      12. 7.12 Symmetries and Killing Vectors
        1. 7.12.1 σ-model Symmetries
        2. 7.12.2 Symmetries of the Poincaré plane
    2. 8 - The First and Second Order Formulations of General Relativity
      1. 8.1 Second Order Formalism for Gravity and Bosonic Matter
      2. 8.2 Gravitational Fluctuations of Flat Spacetime
        1. 8.2.1 The Graviton Green’s Function
      3. 8.3 Second Order Formalism for Gravity and Fermions
      4. 8.4 First Order Formalism for Gravity and Fermions
  10. Part III: Basic Supergravity
    1. 9 - N = 1 Pure Supergravity in Four Dimensions
      1. 9.1 The Universal Part of Supergravity
      2. 9.2 Supergravity in the First Order Formalism
      3. 9.3 The 1.5 Order Formalism
      4. 9.4 Local Supersymmetry of N = 1, D = 4 Supergravity
      5. 9.5 The Algebra of Local Supersymmetry
      6. 9.6 Anti-de Sitter Supergravity
    2. 10 - D = 11 Supergravity
      1. 10.1 D ≤ 11 From Dimensional Reduction
      2. 10.2 The Field Content of D = 11 Supergravity
      3. 10.3 Construction of the Action and Transformation Rules
      4. 10.4 The Algebra of D = 11 Supergravity
    3. 11 - General Gauge Theory
      1. 11.1 Symmetries
        1. 11.1.1 Global Symmetries
        2. 11.1.2 Local Symmetries and Gauge Fields
        3. 11.1.3 Modified Symmetry Algebras
      2. 11.2 Covariant Quantities
        1. 11.2.1 Covariant Derivatives
        2. 11.2.2 Curvatures
      3. 11.3 Gauged Spacetime Translations
        1. 11.3.1 Gauge Transformations for the Poincaré Group
        2. 11.3.2 Covariant Derivatives and General Coordinate Transformations
        3. 11.3.3 Covariant Derivatives and Curvatures in a Gravity Theory
        4. 11.3.4 Calculating Transformations of Covariant Quantities
      4. Appendix 11A Manipulating Covariant Derivatives
        1. 11A.1 Proof of the Main Lemma
        2. 11A.2 Examples in Supergravity
    4. 12 - Survey of Supergravities
      1. 12.1 The Minimal Superalgebras
        1. 12.1.1 Four Dimensions
        2. 12.1.2 Minimal Superalgebras in Higher Dimensions
      2. 12.2 The R-Symmetry Group
      3. 12.3 Multiplets
        1. 12.3.1 Multiplets in Four Dimensions
        2. 12.3.2 Multiplets in More than Four Dimensions
      4. 12.4 Supergravity Theories: Towards a Catalogue
        1. 12.4.1 The Basic Theories and Kinetic Terms
        2. 12.4.2 Deformations and Gauged Supergravities
      5. 12.5 Scalars and Geometry
      6. 12.6 Solutions and Preserved Supersymmetries
        1. 12.6.1 Anti-de Sitter Superalgebras
        2. 12.6.2 Central Charges in Four Dimensions
        3. 12.6.3 ‘Central Charges’ in Higher Dimensions
  11. Part IV: Complex Geometry and Global SUSY
    1. 13 - Complex Manifolds
      1. 13.1 The Local Description of Complex and Kähler Manifolds
      2. 13.2 Mathematical Structure of Kähler Manifolds
      3. 13.3 The Kähler Manifolds CPn
      4. 13.4 Symmetries of Kähler Metrics
        1. 13.4.1 Holomorphic Killing Vectors and Moment Maps
        2. 13.4.2 Algebra of Holomorphic Killing Vectors
        3. 13.4.3 The Killing Vectors of CP1
    2. 14 - General Actions with N = 1 Supersymmetry
      1. 14.1 Multiplets
        1. 14.1.1 Chiral Multiplets
        2. 14.1.2 Real Multiplets
      2. 14.2 Generalized Actions by Multiplet Calculus
        1. 14.2.1 The Superpotential
        2. 14.2.2 Kinetic Terms for Chiral Multiplets
        3. 14.2.3 Kinetic Terms for Gauge Multiplets
      3. 14.3 Kähler Geometry from Chiral Multiplets
      4. 14.4 General Couplings of Chiral Multiplets and Gauge Multiplets
        1. 14.4.1 Global Symmetries of the SUSY σ-model
        2. 14.4.2 Gauge and SUSY Transformations for Chiral Multiplets
        3. 14.4.3 Actions of Chiral Multiplets in a Gauge Theory
        4. 14.4.4 General Kinetic Action of the Gauge Multiplet
        5. 14.4.5 Requirements for an N = 1 SUSY Gauge Theory
      5. 14.5 The Physical Theory
        1. 14.5.1 Elimination of Auxiliary Fields
        2. 14.5.2 The Scalar Potential
        3. 14.5.3 The Vacuum State and SUSY Breaking
        4. 14.5.4 Supersymmetry Breaking and the Goldstone Fermion
        5. 14.5.5 Mass Spectra and the Supertrace Sum Rule
        6. 14.5.6 Coda
      6. Appendix 14A Superspace
      7. Appendix 14B Appendix: Covariant Supersymmetry Transformations
  12. Part V: Superconformal Construction of Supergravity Theories
    1. 15 - Gravity as a Conformal Gauge Theory
      1. 15.1 The Strategy
      2. 15.2 The Conformal Algebra
      3. 15.3 Conformal Transformations on Fields
      4. 15.4 The Gauge Fields and Constraints
      5. 15.5 The Action
      6. 15.6 Recapitulation
      7. 15.7 Homothetic Killing Vectors
    2. 16 - The Conformal Approach to Pure N = 1 Supergravity
      1. 16.1 Ingredients
        1. 16.1.1 Superconformal Algebra
        2. 16.1.2 Gauge fields, Transformations, and Curvatures
        3. 16.1.3 Constraints
        4. 16.1.4 Superconformal Transformation Rules of a Chiral Multiplet
      2. 16.2 The action
        1. 16.2.1 Superconformal Action of the Chiral Multiplet
        2. 16.2.2 Gauge Fixing
        3. 16.2.3 The Result
    3. 17 - Construction of the Matter-Coupled N = 1 Supergravity
      1. 17.1 Superconformal Tensor Calculus
        1. 17.1.1 The Superconformal Gauge Multiplet
        2. 17.1.2 The Superconformal Real Multiplet
        3. 17.1.3 Gauge Transformations of Superconformal Chiral Multiplets
        4. 17.1.4 Invariant actions
      2. 17.2 Construction of the Action
        1. 17.2.1 Conformal Weights
        2. 17.2.2 Superconformal Invariant Action (Ungauged)
        3. 17.2.3 Gauged Superconformal Supergravity
        4. 17.2.4 Elimination of Auxiliary Fields
        5. 17.2.5 Partial Gauge Fixing
      3. 17.3 Projective Kähler Manifolds
        1. 17.3.1 The Example of CPn
        2. 17.3.2 Dilatations and Holomorphic Homothetic Killing Vectors
        3. 17.3.3 The Projective Parametrization
        4. 17.3.4 The Kähler Cone
        5. 17.3.5 The Projection
        6. 17.3.6 Kähler Transformations
        7. 17.3.7 Physical Fermions
        8. 17.3.8 Symmetries of Projective Kähler Manifolds
        9. 17.3.9 T-gauge and Decomposition Laws
        10. 17.3.10 An Explicit Example: SU(1, 1)/U(1) Model
      4. 17.4 From Conformal to Poincaré Supergravity
        1. 17.4.1 The Superpotential
        2. 17.4.2 The Potential
        3. 17.4.3 Fermion Terms
      5. 17.5 Review and Preview
        1. 17.5.1 Projective and Kähler–Hodge manifolds
        2. 17.5.2 Compact Manifolds
      6. Appendix 17A Kähler–Hodge Manifolds
        1. 17A.1 Dirac Quantization Condition
        2. 17A.2 Kähler–Hodge Manifolds
      7. Appendix 17B Steps in the Derivation of (17.7)
  13. Part VI: N = 1 Supergravity Actions and Applications
    1. 18 - The Physical N = 1 Matter-Coupled Supergravity
      1. 18.1 The Physical Action
      2. 18.2 Transformation Rules
      3. 18.3 Further Remarks
        1. 18.3.1 Engineering Dimensions
        2. 18.3.2 Rigid or Global Limit
        3. 18.3.3 Quantum Effects and Global Symmetries
    2. 19 - Applications of N = 1 Supergravity
      1. 19.1 Supersymmetry Breaking and the Super-BEH Effect
        1. 19.1.1 Goldstino and the Super-BEH Effect
        2. 19.1.2 Extension to Cosmological Solutions
        3. 19.1.3 Mass Sum Rules in Supergravity
      2. 19.2 The Gravity Mediation Scenario
        1. 19.2.1 The Polónyi Model of the Hidden Sector
        2. 19.2.2 Soft SUSY breaking in the Observable Sector
      3. 19.3 No-Scale models
      4. 19.4 Supersymmetry and Anti-de Sitter Space
      5. 19.5 R-symmetry and Fayet–Iliopoulos terms
        1. 19.5.1 The R-gauge Field and Transformations
        2. 19.5.2 Fayet–Iliopoulos Terms
        3. 19.5.3 An Example with Non-Minimal Kähler Potential
  14. Part VII: Extended N = 2 Supergravity
    1. 20 - Construction of the Matter-Coupled N = 2 Supergravity
      1. 20.1 Global Supersymmetry
        1. 20.1.1 Gauge Multiplets for D = 6
        2. 20.1.2 Gauge Multiplets for D = 5
        3. 20.1.3 Gauge Multiplets for D = 4
        4. 20.1.4 Hypermultiplets
        5. 20.1.5 Gauged Hypermultiplets
      2. 20.2 N = 2 Superconformal Calculus
        1. 20.2.1 The Superconformal Algebra
        2. 20.2.2 Gauging of the Superconformal Algebra
        3. 20.2.3 Conformal Matter Multiplets
        4. 20.2.4 Superconformal Actions
        5. 20.2.5 Partial Gauge Fixing
        6. 20.2.6 Elimination of Auxiliary Fields
        7. 20.2.7 Complete Action
        8. 20.2.8 D = 5 and D = 6, N = 2 Supergravities
      3. 20.3 Special Geometry
        1. 20.3.1 The Family of Special Manifolds
        2. 20.3.2 Very Special Real Geometry
        3. 20.3.3 Special Kähler Geometry
        4. 20.3.4 Hyper-Kähler and Quaternionic-Kähler Manifolds
      4. 20.4 From Conformal to Poincaré Supergravity
        1. 20.4.1 Kinetic Terms of the Bosons
        2. 20.4.2 Identities of Special Kähler Geometry
        3. 20.4.3 The Potential
        4. 20.4.4 Physical Fermions and Other Terms
        5. 20.4.5 Supersymmetry and Gauge Transformations
      5. Appendix 20A SU(2) Conventions and Triplets
      6. Appendix 20B Dimensional Reduction 6 → 5 → 4
        1. 20B.1 Reducing from D = 6 → D = 5
        2. 20B.2 Reducing from D = 5 → D = 4
      7. Appendix 20C Definition of Rigid Special Kähler Geometry
    2. 21 - The Physical N = 2 Matter-Coupled Supergravity
      1. 21.1 The Bosonic Sector
        1. 21.1.1 The Basic (Ungauged) N = 2, D = 4 Matter-Coupled Supergravity
        2. 21.1.2 The Gauged Supergravities
      2. 21.2 The Symplectic Formulation
        1. 21.2.1 Symplectic Definition
        2. 21.2.2 Comparison of Symplectic and Prepotential Formulation
        3. 21.2.3 Gauge Transformations and Symplectic Vectors
        4. 21.2.4 Physical Fermions and Duality
      3. 21.3 Action and Transformation Laws
        1. 21.3.1 Final Action
        2. 21.3.2 Supersymmetry Transformations
      4. 21.4 Applications
        1. 21.4.1 Partial Supersymmetry Breaking
        2. 21.4.2 Field Strengths and Central Charges
        3. 21.4.3 Moduli Spaces of Calabi–Yau Manifolds
      5. 21.5 Remarks
        1. 21.5.1 Fayet–Iliopoulos Terms
        2. 21.5.2 σ-model Symmetries
        3. 21.5.3 Engineering Dimensions
  15. Part VIII: Classical Solutions and the AdS/CFT Correspondence
    1. 22 - Classical Solutions of Gravity and Supergravity
      1. 22.1 Some Solutions of the Field Equations
        1. 22.1.1 Prelude: Frames and Connections on Spheres
        2. 22.1.2 Anti-de Sitter Space
        3. 22.1.3 AdSD Obtained from its Embedding in ℝD+1
        4. 22.1.4 Space Time Metrics with Spherical Symmetry
        5. 22.1.5 AdS–Schwarzschild Spacetime
        6. 22.1.6 The Reissner–Nordström Metric
        7. 22.1.7 A More General Reissner–Nordström Solution
      2. 22.2 Killing Spinors and BPS Solutions
        1. 22.2.1 The Integrability Condition for Killing Spinors
        2. 22.2.2 Commuting and Anti-Commuting Killing Spinors
      3. 22.3 Killing Spinors for Anti-de Sitter Space
      4. 22.4 Extremal Reissner–Nordström Spacetimes as BPS Solutions
      5. 22.5 The Black Hole Attractor Mechanism
        1. 22.5.1 Example of a Black Hole Attractor
        2. 22.5.2 The Attractor Mechanism – Real Slow and Simple
      6. 22.6 Supersymmetry of the Black Holes
        1. 22.6.1 Killing Spinors
        2. 22.6.2 The Central Charge
        3. 22.6.3 The Black Hole Potential
      7. 22.7 First Order Gradient Flow Equations
      8. 22.8 The Attractor Mechanism – Fast and Furious
      9. Appendix 22A Killing Spinors for pp-waves
    2. 23 - The AdS/CFT Correspondence
      1. 23.1 The N = 4 SYM Theory
      2. 23.2 Type IIB String Theory and D3-Branes
      3. 23.3 The D3-Brane Solution of Type IIB Supergravity
      4. 23.4 Kaluza–Klein Analysis on AdS5 ⊗ S5
      5. 23.5 Euclidean AdS and its Inversion Symmetry
      6. 23.6 Inversion and CFT Correlation Functions
      7. 23.7 The Free Massive Scalar Field in Euclidean AdSd+1
      8. 23.8 AdS/CFT Correlators in a Toy Model
      9. 23.9 Three-Point Correlation Functions
      10. 23.10 Two-Point Correlation Functions
      11. 23.11 Holographic Renormalization
        1. 23.11.1 The Scalar Two-Point Function in a CFTd
        2. 23.11.2 The Holographic Trace Anomaly
      12. 23.12 Holographic RG Flows
        1. 23.12.1 AAdS Domain Wall Solutions
        2. 23.12.2 The Holographic c-theorem
        3. 23.12.3 First Order Flow Equations
      13. 23.13 AdS/CFT and Hydrodynamics
  16. Appendix A - Comparison of Notation
    1. A.1 Spacetime and Gravity
    2. A.2 Spinor Conventions
    3. A.3 Components of Differential Forms
    4. A.4 Covariant Derivatives
  17. Appendix B - Lie Algebras and Superalgebras
    1. B.1 Groups and Representations
    2. B.2 Lie Algebras
    3. B.3 Superalgebras
  18. References
  19. Index