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Introduction to Strings and Branes

Book Description

Supersymmetry, strings and branes are believed to be the essential ingredients in a single unified consistent theory of physics. This book gives a detailed, step-by-step introduction to the theoretical foundations required for research in strings and branes. After a study of the different formulations of the bosonic and supersymmetric point particles, the classical and quantum bosonic and supersymmetric string theories are presented. This book includes accounts of brane dynamics and D-branes and the T, S and U duality symmetries of string theory. The historical derivation of string theory is given as well as the sum over the world-sheet approach to the interacting string. More advanced topics include string field theory and Kac-Moody symmetries. The book contains pedagogical accounts of conformal quantum field theory, supergravity theories, Clifford algebras and spinors, and Lie algebras. It is essential reading for graduate students and researchers wanting to learn strings and branes.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. 1 The point particle
    1. 1.1 The bosonic point particle
      1. 1.1.1 The classical point particle and its Dirac quantisation
      2. 1.1.2 The BRST quantization of the point particle
    2. 1.2 The super point particle
      1. 1.2.1 The spinning particle
      2. 1.2.2 The Brink–Schwarz superparticle
      3. 1.2.3 Superspace formulation of the point particle
    3. 1.3 The twistor approach to the massless point particle
      1. 1.3.1 Twistors in four and three dimensions
      2. 1.3.2 The twistor point particle actions
  7. 2 The classical bosonic string
    1. 2.1 The dynamics
      1. 2.1.1 The closed string
      2. 2.1.2 The open string
    2. 2.2 The energy-momentum and angular momentum of the string
    3. 2.3 A classical solution of the open string
  8. 3 The quantum bosonic string
    1. 3.1 The old covariant method
      1. 3.1.1 The open string
      2. 3.1.2 The closed string
    2. 3.2 The BRST approach
      1. 3.2.1 The BRST action
      2. 3.2.2 The world-sheet energy-momentum tensor and BRST charge
      3. 3.2.3 The physical state condition
  9. 4 The light-cone approach
    1. 4.1 The classical string in the light-cone
    2. 4.2 The quantum string in the light-cone
    3. 4.3 Lorentz symmetry
    4. 4.4 Light-cone string field theory
  10. 5 Clifford algebras and spinors
    1. 5.1 Clifford algebras
    2. 5.2 Clifford algebras in even dimensions
    3. 5.3 Spinors in even dimensions
    4. 5.4 Clifford algebras in odd dimensions
    5. 5.5 Central charges
    6. 5.6 Clifford algebras in space-times of arbitrary signature
  11. 6 The classical superstring
    1. 6.1 The Neveu–Schwarz–Ramond (NS–R) formulation
      1. 6.1.1 The open superstring
      2. 6.1.2 The closed superstring
    2. 6.2 The Green–Schwarz formulation
  12. 7 The quantum superstring
    1. 7.1 The old covariant approach to the open superstring
      1. 7.1.1 The NS sector
      2. 7.1.2 The R sector
    2. 7.2 The GSO projector for the open string
    3. 7.3 The old covariant approach to the closed superstring
  13. 8 Conformal symmetry and two-dimensional field theory
    1. 8.1 Conformal transformations
      1. 8.1.1 Conformal transformations in D dimensions
      2. 8.1.2 Conformal transformations in two dimensions
    2. 8.2 Conformally invariant two-dimensional field theories
      1. 8.2.1 Conformally invariant two-dimensional classical theories
      2. 8.2.2 Conformal Ward identities
    3. 8.3 Constraints due to global conformal transformations
    4. 8.4 Transformations of the energy-momentum tensor
    5. 8.5 Operator product expansions
    6. 8.6 Commutators
    7. 8.7 Descendants
    8. 8.8 States, modes and primary fields
    9. 8.9 Representations of the Virasoro algebra and minimal models
  14. 9 Conformal symmetry and string theory
    1. 9.1 Free field theories
      1. 9.1.1 The free scalar
      2. 9.1.2 The free fermion
    2. 9.2 First order systems
    3. 9.3 Application to string theory
      1. 9.3.1 Mapping the string to the Riemann sphere
      2. 9.3.2 Construction of string theories
    4. 9.4 The free field representation of the minimal models
  15. 10 String compactification and the heterotic string
    1. 10.1 Compactification on a circle
    2. 10.2 Torus compactification
    3. 10.3 Compactification in the presence of background fields
    4. 10.4 Description of the moduli space
    5. 10.5 Heterotic compactification
    6. 10.6 The heterotic string
  16. 11 The physical states and the no-ghost theorem
    1. 11.1 The no-ghost theorem
    2. 11.2 The zero-norm physical states
    3. 11.3 The physical state projector
    4. 11.4 The cohomology of Q
  17. 12 Gauge covariant string theory
    1. 12.1 The problem
    2. 12.2 The solution
    3. 12.3 Derivation of the solution
    4. 12.4 The gauge covariant closed string
    5. 12.5 The gauge covariant superstring
  18. 13 Supergravity theories in four, ten and eleven dimensions
    1. 13.1 Four ways to construct supergravity theories
      1. 13.1.1 The Noether method
      2. 13.1.2 The on-shell superspace method
      3. 13.1.3 Gauging of space-time groups
      4. 13.1.4 Dimensional reduction
    2. 13.2 Non-linear realisations
    3. 13.3 Eleven-dimensional supergravity
    4. 13.4 The IIA supergravity theory
    5. 13.5 The IIB supergravity theory
      1. 13.5.1 The algebra and field content
      2. 13.5.2 The equations of motion
      3. 13.5.3 The SL(2,R) version
    6. 13.6 Symmetries of the maximal supergravity theories in dimensions less than ten
    7. 13.7 Type I supergravity and supersymmetric Yang–Mills theories in ten dimensions
    8. 13.8 Solutions of the supergravity theories
      1. 13.8.1 Solutions in a generic theory
      2. 13.8.2 Brane solutions in eleven-dimensional supergravity
      3. 13.8.3 Brane solutions in the ten-dimensional maximal supergravity theories
      4. 13.8.4 Brane charges and the preservation of supersymmetry
  19. 14 Brane dynamics
    1. 14.1 Bosonic branes
    2. 14.2 Types of superbranes
    3. 14.3 Simple superbranes
    4. 14.4 D-branes
    5. 14.5 Branes in M theory
    6. 14.6 Solutions of the 5-brane of M theory
      1. 14.6.1 The 3-brane
      2. 14.6.2 The self-dual string
    7. 14.7 Five-brane dynamics and the low energy effective action of the N = 2 Yang–Mills theory
  20. 15 D-branes
    1. 15.1 Bosonic D-branes
    2. 15.2 Super D-branes in the NS–R formulation
    3. 15.3 D-branes in the Green–Schwarz formulation
  21. 16 String theory and Lie algebras
    1. 16.1 Finite dimensional and affine Lie algebras
      1. 16.1.1 A review of finite-dimensional Lie algebras and lattices
      2. 16.1.2 Representations of finite dimensional semi-simple Lie algebras
      3. 16.1.3 Affine Lie algebras
    2. 16.2 Kac–Moody algebras
    3. 16.3 Lorentzian algebras
    4. 16.4 Very extended and over-extended Lie algebras
    5. 16.5 Weights and inverse Cartan matrix of En
    6. 16.6 Low level analysis of Lorentzian Kac–Moody algebras
      1. 16.6.1 The adjoint representation
      2. 16.6.2 All representations
    7. 16.7 The Kac–Moody algebra E11
      1. 16.7.1 E11 at low levels
      2. 16.7.2 The l1 representation of E11
      3. 16.7.3 The Cartan involution invariant subalgebra of a Kac–Moody algebra
    8. 16.8 String vertex operators and Lie algebras
  22. 17 Symmetries of string theory
    1. 17.1 T duality
    2. 17.2 Electromagnetic duality
    3. 17.3 S and U duality
    4. 17.4 M theory
    5. 17.5 E theory
      1. 17.5.1 The eleven-dimensional theory
      2. 17.5.2 The IIA and IIB theories
      3. 17.5.3 The common origin of the eleven-dimensional, IIA and IIB theories
      4. 17.5.4 Theories in less than ten dimensions
      5. 17.5.5 Duality symmetries and conditions
      6. 17.5.6 Brane charges, the l1 representation and generalised space-time
      7. 17.5.7 Weyl transformations of E11 and the non-linear realisation of its Cartan sub-algebra
  23. 18 String interactions
    1. 18.1 Duality, factorisation and the origins of string theory
    2. 18.2 The path integral approach
    3. 18.3 The group theoretic approach
    4. 18.4 Interacting open string field theory
      1. 18.4.1 Light-cone string field theory
      2. 18.4.2 Mapping the interacting string
      3. 18.4.3 A brief discussion of interacting gauge covariant string field theory
  24. Appendix A The Dirac and BRST methods of quantisation
    1. A.1 The Dirac method
    2. A.2 The BRST method
  25. Appendix B Two-dimensional light-cone and spinor conventions
    1. B.1 Light-cone coordinates
    2. B.2 Spinor conventions
  26. Appendix C The relationship between S2 and the Riemann sphere
  27. Appendix D Some properties of the classical Lie algebras
    1. D.1 The algebras An−1
    2. D.2 The algebras Dn
    3. D.3 The algebra E6
    4. D.4 The algebra E7
    5. D.5 The algebra E8
    6. D.6 The algebras Bn
    7. D.7 The algebras Cn
  28. Chapter quote acknowledgements
  29. References
  30. Index