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Superstring Theory

Book Description

Twenty-five years ago, Michael Green, John Schwarz, and Edward Witten wrote two volumes on string theory. Published during a period of rapid progress in this subject, these volumes were highly influential for a generation of students and researchers. Despite the immense progress that has been made in the field since then, the systematic exposition of the foundations of superstring theory presented in these volumes is just as relevant today as when first published. Volume 2 is concerned with the evaluation of one-loop amplitudes, the study of anomalies and phenomenology. It examines the low energy effective field theory analysis of anomalies, the emergence of the gauge groups E8 x E8 and SO(32) and the four-dimensional physics that arises by compactification of six extra dimensions. Featuring a new Preface setting the work in context in light of recent advances, this book is invaluable for graduate students and researchers in high energy physics and astrophysics, as well as mathematicians.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface to the 25th Anniversary Edition
  6. 8 One-loop diagrams in the bosonic string theory
    1. 8.1 Open-string one-loop amplitudes
      1. 8.1.1 The planar diagrams
      2. 8.1.2 The nonorientable diagrams
      3. 8.1.3 Nonplanar loop diagrams
    2. 8.2 Closed-string one-loop amplitudes
      1. 8.2.1 The torus
      2. 8.2.2 Modular invariance
      3. 8.2.3 The integration region
      4. 8.2.4 Analysis of divergences
      5. 8.2.5 The cosmological constant
      6. 8.2.6 Amplitudes with closed-string massless states
    3. 8.3 Other diagrams for unoriented strings
      1. 8.3.1 Higher-order tree diagrams
      2. 8.3.2 The real projective plane
      3. 8.3.3 Other loop diagrams
    4. 8.4 Summary
    5. 8.A Jacobi Θ functions
  7. 9 One-loop diagrams in superstring theory
    1. 9.1 Open-superstring amplitudes
      1. 9.1.1 Amplitudes with M < 4 massless external states
      2. 9.1.2 The planar diagrams
      3. 9.1.3 Nonorientable diagrams
      4. 9.1.4 Orientable nonplanar diagrams
    2. 9.2 Type II theories
      1. 9.2.1 Finiteness of the torus amplitude
      2. 9.2.2 Compactification on a torus
      3. 9.2.3 The low-energy limit of one-loop amplitudes
    3. 9.3 The heterotic string theory
      1. 9.3.1 The torus with four external particles
      2. 9.3.2 Modular invariance of the E8 × E8 and SO(32) theories
    4. 9.4 Calculations in the RNS formalism
      1. 9.4.1 Modular invariance and the GSO projection
      2. 9.4.2 The loop calculations
    5. 9.5 Orbifolds and twisted strings
      1. 9.5.1 Generalization of the GSO projection
      2. 9.5.2 Strings on orbifolds
      3. 9.5.3 Twisted strings in ten dimensions
      4. 9.5.4 Alternative view of the SO(16) × SO(16) theory
    6. 9.6 Summary
    7. 9.A Traces of fermionic zero modes
    8. 9.B Modular invariance of the functions F2 and L
  8. 10 The gauge anomaly in type I superstring theory
    1. 10.1 Introduction to anomalies
      1. 10.1.1 Anomalies in point-particle field theory
      2. 10.1.2 The gauge anomaly in D = 10 super Yang–Mills theory
      3. 10.1.3 Anomalies in superstring theory
    2. 10.2 Analysis of hexagon diagrams
      1. 10.2.1 The planar diagram anomaly
      2. 10.2.2 The anomaly in the nonorientable diagram
      3. 10.2.3 Absence of anomalies in nonplanar diagrams
    3. 10.3 Other one-loop anomalies in superstring theory
    4. 10.4 Cancellation of divergences for SO(32)
      1. 10.4.1 Dilaton tadpoles and loop divergences
      2. 10.4.2 Divergence cancellations
      3. 10.5 Summary
    5. 10.A An alternative regulator
  9. 11 Functional methods in the light-cone gauge
    1. 11.1 The string path integral
      1. 11.1.1 The analog model
      2. 11.1.2 The free string propagator
      3. 11.1.3 A lattice cutoff
      4. 11.1.4 The continuum limit
    2. 11.2 Amplitude calculations
      1. 11.2.1 Interaction vertices
      2. 11.2.2 Parametrization of scattering processes
      3. 11.2.3 Evaluation of the functional integral
      4. 11.2.4 Amplitudes with external ground states
    3. 11.3 Open-string tree amplitudes
      1. 11.3.1 The conformal mapping
      2. 11.3.2 Evaluation of amplitudes
    4. 11.4 Open-string trees with excited external states
      1. 11.4.1 The Green function on an infinite strip
      2. 11.4.2 Green functions for arbitrary tree amplitudes
      3. 11.4.3 The amplitude in terms of oscillators
      4. 11.4.4 The general form of the Neumann coefficients
      5. 11.4.5 The Neumann coefficients for the cubic open-string vertex
    5. 11.5 One-loop open-string amplitudes
      1. 11.5.1 The conformal mapping for the planar loop diagram
      2. 11.5.2 The Green function
      3. 11.5.3 The planar one-loop amplitude
      4. 11.5.4 Other one-loop amplitudes
    6. 11.6 Closed-string amplitudes
      1. 11.6.1 Tree amplitudes
      2. 11.6.2 Closed-string one-loop amplitudes
    7. 11.7 Superstrings
      1. 11.7.1 The SU(4) × U(1) formalism
      2. 11.7.2 The super-Poincaré generators
      3. 11.7.3 Supersymmetry algebra in the interacting theory
      4. 11.7.4 The continuity delta functional
      5. 11.7.5 Singular operators near the interaction point
      6. 11.7.6 The interaction terms
      7. 11.7.7 Tree amplitudes for open superstrings
    8. 11.8 Summary
    9. 11.A The determinant of the Laplacian
    10. 11.B The Jacobian for the conformal transformation
    11. 11.C Properties of the functions fm
    12. 11.D Properties of the SU(4) Clebsch-Gordan coefficients
  10. 12 Some differential geometry
    1. 12.1 Spinors in general relativity
    2. 12.2 Spin structures on the string world sheet
    3. 12.3 Topologically nontrivial gauge fields
      1. 12.3.1 The tangent bundle
      2. 12.3.2 Gauge fields and vector bundles
    4. 12.4 Differential forms
    5. 12.5 Characteristic classes
      1. 12.5.1 The nonabelian case
      2. 12.5.2 Characteristic classes of manifolds
      3. 12.5.3 The Euler characteristic of a Riemann surface
  11. 13 Low-energy effective action
    1. 13.1 Minimal supergravity plus super Yang–Mills
      1. 13.1.1 N = 1 supergravity in ten and eleven dimensions
      2. 13.1.2 Type IIB supergravity
      3. 13.1.3 The coupled supergravity super Yang–Mills system
    2. 13.2 Scale invariance of the classical theory
    3. 13.3 Anomaly analysis
      1. 13.3.1 Structure of field theory anomalies
      2. 13.3.2 Gravitational anomalies
      3. 13.3.3 Mixed anomalies
      4. 13.3.4 The anomalous Feynman diagrams
      5. 13.3.5 Mathematical characterization of anomalies
      6. 13.3.6 Other types of anomalies
    4. 13.4 Explicit formulas for the anomalies
    5. 13.5 Anomaly cancellations
      1. 13.5.1 Type I supergravity without matter
      2. 13.5.2 Type IIB supergravity
      3. 13.5.3 Allowed gauge groups for N = 1 superstring theories
      4. 13.5.4 The SO(16) × SO(16) theory
  12. 14 Compactification of higher dimensions
    1. 14.1 Wave operators in ten dimensions
      1. 14.1.1 Massless fields in ten dimensions
      2. 14.1.2 Zero modes of wave operators
    2. 14.2 Massless fermions
      1. 14.2.1 The index of the Dirac operator
      2. 14.2.2 Incorporation of gauge fields
      3. 14.2.3 The chiral asymmetry
      4. 14.2.4 The Rarita-Schwinger operator
      5. 14.2.5 Outlook
    3. 14.3 Zero modes of antisymmetric tensor fields
      1. 14.3.1 Antisymmetric tensor fields
      2. 14.3.2 Application to axions in N = 1 superstring theory
      3. 14.3.3 The ‘nonzero modes’
      4. 14.3.4 The exterior derivative and the Dirac operator
    4. 14.4 Index theorems on the string world sheet
      1. 14.4.1 The Dirac index
      2. 14.4.2 The Euler characteristic
      3. 14.4.3 Zero modes of conformal ghosts
      4. 14.4.4 Zero modes of superconformal ghosts
    5. 14.5 Zero modes of nonlinear fields
    6. 14.6 Models of the fermion quantum numbers
    7. 14.7 Anomaly cancellation in four dimensions
  13. 15 Some algebraic geometry
    1. 15.1 Low-energy supersymmetry
      1. 15.1.1 Motivation
      2. 15.1.2 Conditions for unbroken supersymmetry
      3. 15.1.3 Manifolds of SU(3) holonomy
    2. 15.2 Complex manifolds
      1. 15.2.1 Almost complex structure
      2. 15.2.2 The Nijenhuis tensor
      3. 15.2.3 Examples of complex manifolds
    3. 15.3 Kähler manifolds
      1. 15.3.1 The Kähler metric
      2. 15.3.2 Exterior derivatives
      3. 15.3.3 The affine connection and the Riemann tensor
      4. 15.3.4 Examples of Kahler manifolds
    4. 15.4 Ricci-flat Kähler manifolds and SU(N) holonomy
      1. 15.4.1 The Calabi–Yau metric
      2. 15.4.2 Covariantly constant forms
      3. 15.4.3 Some manifolds of SU(N) holonomy
    5. 15.5 Wave operators on Kahler manifolds
      1. 15.5.1 The Dirac operator
      2. 15.5.2 Dolbeault cohomology
      3. 15.5.3 The Hodge decomposition
      4. 15.5.4 Hodge numbers
    6. 15.6 Yang–Mills equations and holomorphic vector bundles
      1. 15.6.1 Holomorphic vector bundles
      2. 15.6.2 The Donaldson–Uhlenbeck–Yau equation
      3. 15.6.3 Examples
    7. 15.7 Dolbeault cohomology and some applications
      1. 15.7.1 Zero modes of the Dirac operator
      2. 15.7.2 Deformations of complex manifolds
      3. 15.7.3 Deformations of holomorphic vector bundles
    8. 15.8 Branched coverings of complex manifolds
  14. 16 Models of low-energy supersymmetry
    1. 16.1 A simple Ansatz
    2. 16.2 The spectrum of massless particles
      1. 16.2.1 Zero modes of charged fields
      2. 16.2.2 Fluctuations of the gravitational field
      3. 16.2.3 The other Bose fields
    3. 16.3 Symmetry breaking by Wilson lines
      1. 16.3.1 Symmetry breaking patterns
      2. 16.3.2 A four generation model
    4. 16.4 Relation to conventional grand unification
      1. 16.4.1 Alternative description of symmetry breaking
      2. 16.4.2 E6 relations among coupling constants
      3. 16.4.3 Counting massless particles
      4. 16.4.4 Fractional electric charges
      5. 16.4.5 Discussion
    5. 16.5 Global symmetries
      1. 16.5.1 CP conservation in superstring models
      2. 16.5.2 R transformations in superstring models
      3. 16.5.3 Global symmetries of the toy model
      4. 16.5.4 Transformation laws of matter fields
    6. 16.6 Topological formulas for Yukawa couplings
      1. 16.6.1 A topological formula for the superpotential
      2. 16.6.2 The kinetic terms
      3. 16.6.3 A nonrenormalization theorem and its consequences
      4. 16.6.4 Application to the toy model
    7. 16.7 Another approach to symmetry breaking
    8. 16.8 Discussion
    9. 16.9 Renormalization of coupling constants
    10. 16.10 Orbifolds and algebraic geometry
    11. 16.11 Outlook
  15. Bibliography
  16. Index