Recent years have brought a revival of work on string theory, which has been a source of fascination since its origins nearly twenty years ago.There seems to be a widely perceived need for a systematic, pedagogical exposition of the present state of knowledge about string theory. We hope that this book will help to meet this need. To give a comprehensive account of such a vast topic as string theory would scarcely be possible,even in two volumes with the length to which these have grown. Indeed,we have had to omit many important subjects, while treating others only sketchily. String field theory is omitted entirely (though the subject of chapter 11 is closely related to light-cone string field theory). Conformal field theory is not developed systematically, though much of the background material needed to understand recent papers on this subject is presented in chapter 3 and elsewhere.

Preface

8  One-loop diagrams in the bosonic string theory

8.1  Open-string one-loop amplitudes

8.1.1 The planar diagrams

8.1.2 The nonorientable diagrams

8.1.3 Nonplanar loop diagrams

8.2  Closed-string one-loop amplitudes

8.2.1 The torus

8.2.2 Modular invariance

8.2.3 The integration region

8.2.4 Analysis of divergences

8.2.5 The cosmological constant

8.2.6 Amplitudes with closed-string massless states

8.3  Other diagrams for unoriented strings

8.3.1 Higher-order tree diagrams

8.3.2   The real projective plane

8.3.3 Other loop diagrams

8.4  Summary

8.A Jacobi 0 functions

9  One-laop diagrams in superstring theory

9.1  Opemsuperstring amplitudes

9.1.1 Amplitudes with M 4 massless external states

9.1.2 The planar diagrams

9.1.3 Nonorientable diagrams

9.1.4 Orientable nonplanar diagrams

9.2  Type II theories

9.2.1 a Finiteness of the torus amplitude

9.2.2 Compactification on a torus

9.2.3 The low-energy limit of one-loop amplitudes

9.3  The hea;erotic string theory

9.3.1 The torus with four external particles

9.3.2 Modular   invariance of the Es a Es and SO(32) theories

9.4  Calculations in the RNS formalism

9.4.1 Modular invariance and the GSO projection

9.4.2 The loop calculations

9.5  Orbifolds and twisted strings

9.5.1 Generalization of the GSO projection

9.5.2 Strings on orbifolds

9.5.3 Twisted strings in ten dimensions

9.5.4 Alternative view of the SO(16) a SO(16) theory

9.6  Summary

9.A  Traces of fermionic zero modes

9.B  Modular invariance of the functions F2 and/:

10  The gauge anomaly in type I superstring theory

10.1 Introduction to anomalies

10.1.1 Anomalies in point-particle field theory

10.1.2 The gauge anomaly in D = 10 super

Yang-Mills theory

10.1.3 Anomalies in superstring theory

10.2 Analysis of hexagon diagrams

10.2.1 The planar diagram anomaly

10.2.2 The anomaly in the nonorientable diagram

10.2.3 Absence of anomalies in nonplanar diagrams

10.3  Other one-loop anomalies in superstring theory

10.4 Cancellation of divergences for SO(32)

10.4.1 Dilaton tadpoles and loop divergences

10.4.2 Divergence cancellations

10.5 Summary

10.A An alternative regulator

11  Functional methods in the light-cone gauge

11.1  The string path integral

11.1.1 The analog model

11.1.2 The free string propagator

11.1.3 A lattice cutoff

11.1.4 The continuum limit

11.2 Amplitude calculations

11.2.1 Interaction vertices

11.2.2 Parametrization of scattering processes

11.2.3 Evaluation of the functional integral

11.2.4 Amplitudes with external ground statm

11.3 Open-string tree amplitudes

11.3.1 The conformal mapping

11.3.2 Evaluation of amplitudes

11.4 Open-string trees with excited external states

11.4.1 The Green function on an infinite strip

11.4.2 Green functions for arbitrary tree amplitudes

11.4.3 The amplitude in terms of oscillators

11.4.4 The general form of the Neumann coefficients

11.4.5 The Neumann coefficients for

the cubic open-string vertex

11.5 One-loop open-string amplitudes

11.5.1 The conformal mapping for the planar loop diagram

11.5.2 The Green function

11.5.3 The planar one-loop amplitude

11.5.4 Other one-loop amplitudes

11.6 Closed-string amplitudes

11.6.1 Tree amplitudes

11.6.2 Closed-string one-loop amplitudes

11.7 Superstrings

11.7.1 The SU(4) a U(1) formalism

11.7.2 The super-Poincara generators

11.7.3 Supersymmetry algebra in the interacting theory

11.7.4 The continuity delta functional

11.7.5 Singular operators near the interaction point

11.7.6 The interaction terms

11.7.7 Tree amplitudes for open superstrings

11.8  Summary

11.A The determinant of the Laplacian

11.B The Jacobian for the conformal transformation

11.C Properties of the functions f  n

11.D Properties of the SU(4) Clebsch-Gordan coefficients

12  Some differential geometry

12.1 Spinors in general relativity

12.2  Spin structures on the string world sheet

12.3 Topologically nontrivial gauge fields

12.3.1 The tangent bundle

12.3.2 Gauge fields and vector bundles

12.4 Differential forms

12.5 Characteristic classes

12.5.1 The nonabelian case

12.5.2 Characteristic closes of manifolds

12.5.3 The Euler characteristic:of a Riemann'arface

13  Low-energy effective action

13.1  Minimal supergravity plus super Yang-Mills

13.1.1 N - 1 supergravity in ten and eleven dimensions

13.1.2 Type IIB supergravity

13.1.3 The coupled supergravity super Yang-Mills system

13.2 Scale invariance of the classical theory

13.3 Anomaly analysis

13.3.1 Structure of field theory anomalies

13.3.2 Gravitational anomalies

13.3.3 Mixed anomalies

13.3.4 The anomalous Feynman diagrams

13.3.5 Mathematical characterization of anomalies

13.3.6 Other types of anomalies

13.4 Explicit formulas for the anomalies

13.5 Anomaly cancellations

13.5.1 Type I supergravity without matter

13.5.2 Type IIB supergravity

13.5.3 Allowed gauge groups for N -- 1 superstring theories

13.5.4 The SO(16) x SO(16) theory

14  Compactiflcation of higher dimensions

14.1 Wave operators in ten dimensions

14.1.1 Massless fields in ten dimensions

14.1.2 Zero modes of wave operators

14.2 Massless fermions

14.2.1 The index of the Dirac operator

14.2.2 Incorporation of gauge fields

14.2.3 The chiral asymmetry

14.2.4 The Parita-Schwinger operator

14.2.5 Outlook

14.3 Zero modes of antisymmetric tensor fields

14.3.1 Antisymmetric tensor fields

14.3.2 Application to axions in N = 1 superstring theory

14.3.3 The 'nonzero modes'

14.3.4 The exterior derivative and the Dirac operator

14.4 Index theorems on the string world sheet

14.4.1 The Dirac index

14.4.2 The Euler characteristic

14.4.3 Zero modes of conformal ghosts

14.4.4 Zero modes of superconformal ghosts

14.5  Zero modes of nonlinear fields

14.6 Models of the fermion quantum numbers

14.7 Anomaly cancellation in four dimensions

15  Some algebraic geometry

15.1 Low-energy supersymmetry

15.1.1 Motivation

15.1.2 Conditions for unbroken supersymmetry

15.1.3 Manifolds of SU(3) holonomy

15.2 Complex manifolds

15.2.1 Almost complex structure

15.2.2 The Nijenhuis tensor

15.2.3 Examples of complex manifolds

15.3  KS   hler manifolds

15.3.1 The Kahler metric

15.3.2 Exterior derivatives

15.3.3 The affine connection and the Riemann tensor

15.3.4 Examples of Kahler manifolds

15.4 Ricci-flat Kahler manifolds and SU(N) halonomy

15.4.1 The Calabi-Yau metric

15.4.2 Covariantly constant forms

15.4.3 Some manifolds of SU(N) holonomy

15.5  Wave operators on Kahler manifolds

15.5.1 The Dirac operator

15.5.2 Dolbeault cohomology

15.5.3 The Hodge decomposition

15.5.4 Hodge numbers

15.6  Yang-Mills equations and holomorphic vector bundles

15.6.1 Holomorphic vector bundles

15.6.2 The Donaldson-Uhlenbeck-Yau equation

15.6.3 Examples

15.7  Dolbeault cohomology and some applications

15.7.1 Zero modes of the Dirac operator

15.7.2 Deformations of complex manifolds

15.7.3 Deformations of holomorphic vector bundles

15.8  Branched coverings of complex manifolds

16  Models of low-energy supersymmetry

16.1 A'simple Ansatz

16.2 The spectrum of massless particles

16.2.1 Zero modes of charged fields

16.2.2 Fluctuations of the gravitational field

16.2.3 The other Bose fields

16.3  Symmetry breaking by Wilson lines

16.3.1 Symmetry breaking patterns

16.3.2 A four generation model

16.4  Relation to conventional grand unification

16.4.1 Alternative description of symmetry breaking

16.4.2 Ee relations among coupling constants

16.4.3 Counting massless particles

16.4.4 Fractional electric charges

16.4.5 Discussion

16.5 Global symmetries

16.5.1 CP conservation in superstring models

16.5.2 R transformations in superstring models

16.5.3 Global symmetries of the toy model

16.5.4 Transformation laws of matter fields

16.6 Topological formulas for Yukawa couplings

16.6.1 A topological formula for the superpotential

16.6.2 The kinetic terms

16.6.3 A nonrenormalization theorem and its consequences

16.6.4 Application to the toy model

16.7 Another approach to symmetry breaking

16.8 Discussion

16.9 Renormalization of coupling constants

16.10 Orbifolds and algebraic geometry

16.11 Outlook

Bibliography

Index