Supergravity and Superstrings: A Geometric Perspective

The following sections are included:

• DEDICATION

• PREFACE

• CONTENTS

In this part we are going to discuss mathematics and, in particular, differential geometry. The chosen topics are classical and can be found in many excellent textbooks, although it may be difficult to find one where they are all collected together and treated at the same elementary level as we treat them here. In spite of this, the spirit underlying our presentation and the choice of the topics is non conventional and it is physical in its nature…

The following sections are included:

• Exterior forms on vector spaces

• Mappings and operations on forms

• Differentiable manifolds, vector fields and differential forms
  • Differentiable manifolds

• Functions, vector fields and differential forms
  • Functions
  • Vector fields
  • Induced mappings on tangent spaces.
  • l-forms on a manifold
  • k-forms on manifolds

• Exterior differentiation and behaviour under mappings
  • Behaviour of k-forms under mappings.

• The Vielbein basis
  • Hodge-duality

• Lie derivative, coordinate transformations and invariance

• APPENDIX Appendix: the δ operator and the Hodge decomposition Inner product
  • Adjoint of the exterior derivative
  • Laplacian
  • Hodge's theorem

The following sections are included:

• Introduction

• Geometry of the linear spaces

• The geometry of general Riemannian manifolds in the vielbein basis

• Relation with the standard world-tensor formalism

The following sections are included:

• Introduction

• Lie groups as manifolds: left and right invariant vector fields

• Maurer-Cartan equations

• Adjoint representation and Killing metric

• Killing metric

• Riemannian geometry of semisimple groups

• Soft group manifolds
  • Soft forms and curvatures
  • Lie derivative on soft group manifolds
  • Horizontality and factorization

• The example of Poincaré and anti de Sitter soft group manifold
  • De Sitter curvatures
  • Poincaré group curvatures
  • Fiber bundle structure

The following sections are included:

• Poincaré Gravity

• Extension to the soft group manifold

• Building rules for the gravity Lagrangians

• Gravity in de Sitter and anti de Sitter space

The following sections are included:

• Geometrical Lagrangian for scalar fields on a rigid background

• Extension to the Poincaré group manifold and interpretation of the Lorentz transformation rules as variational equations

• The interaction of the scalar fields with gravity and the effective cosmological constant

• The field equation of a massless scalar in anti de Sitter space (in general in a curved space)

• Geometrical Lagrangians for spin 1 fields

• Geometrical Lagrangian for spin 1/2 fields

The following sections are included:

• Introduction

• Classification of coset manifolds
  • Reductive G/H

• Coordinates on G/H and finite G-transformation
  • Compact G/H
  • Noncompact G/H

• Finite transformations on G/H

• Infinitesimal transformations and Killing vectors

• Vielbeins and metric on G/H

• Covariant Lie derivative

• Geodesics

• Invariant measure

• Connection and curvature
  • Structure constants

• Rescalings

• A note on the isometries of G/H
  • Symmetric rescalings

• Some examples
  • The M^pqr spaces
  • Topology and symmetries
  • Rescaled curvatures
  • Einstein metrics on M^pqr

• Elements of algebraic topology
  • De Rahm cohomology

• Homotopy and (co)homology of coset spaces
  • Homotopy
  • Homology

The following sections are included:

• The Brans-Dicke theory

• Minimal coupling of pseudoscalars through a torsion mechanism

• The Schwarzschild solution

• BIBLIOGRAPHY

In Part One we studied the Einstein theory of gravitation and we elucidated its differential geometric structure, which is best understood and most clear in the language of exterior forms. These latter, on the other hand, are naturally associated to the dual formulation of Lie algebras via Maurer Cartan equations and to their soft deformations. The Lie algebra one deals with in gravity is that of the Poincaré group (or (anti) de-Sitter group) which can be viewed as the group of motions of flat Minkowski space (or (anti) de Sitter space), namely the vacuum of gravity theory…

The following sections are included:

• The definition of superalgebras and the example of N-extended super Poincare algebra

• Classification of the simple superalgebras whose Lie algebra is reductive
  • The orthosymplectic algebras 0sp(2p/N)
  • The superunitary algebras SU(p,q/N)
  • The superalgebras P(n) and Q(n)
  • The exceptional superalgebras D(2,1,α), G(3), F(4)

• Grassmann algebras

• Supermanifolds

• Supergroups and graded matrices

• 0sp(4/N) as the N-extended supersymmetry algebra in anti de Sitter space

The following sections are included:

• Maurer-Cartan equations of supergroups on supergroup manifolds

• Maurer-Cartan equations of 0sp(4/N) and

• 0sp(4/N) Maurer-Cartan equations as the structure equations of rigid superspace

• Killing vectors on superspace, that is the generators of the supersymmetry algebra of superisometries

• Minkowski N-extended superspace

The following sections are included:

• How to construct the unitary irreducible representations of the N-extended Poincaré superalgebra
  • N=l Supersymmetry

• Massive multiplets without central charges

• Massive Multiplets with Central Charges

• Massless Multiplets

The following sections are included:

• Free field equations and the concept of mass in anti de Sitter space
  • The s=0 particle
  • The s=1/2 particle
  • The spin 1 particle
  • The spin 3/2 particle
  • The spin 2 particle

• Unitary irreducible representations of SO(2,3)

• Unitary irreducible representations of 0sp(4/N)

• 0sp(4/1) supermultiplets

• Remarks about the N-extended case and the example of the 0sp(4/2) multiplets

The following sections are included:

• Supersymmetric field-theories corresponding to an irreducible representation of the supersymmetry algebra

• The Wess-Zumino model: the simplest example of a supersymmetric field theory

• Superfield interpretation of the Wess-Zumino model and rheonomy

• The integrability of the rheonomic conditions and the Bianchi identities

• The rheonomic action principle

The following sections are included:

• The construction of Г-matrices
  • D 2v= even
  • D 2v+1 = odd

• The charge conjugation matrix

• Majorana, Weyl and Majorana-Weyl spinors

• Useful formulae in Г-matrix algebra

The following sections are included:

• Introduction

• The structure of forms on N-extended D=4 superspace

• Fierz decompositions in the N=l D=4 superspace

• The N=2, D=4 case

• The N=3, D=4 case

• The N=2,D=5 case

• Systematics of Fierz identities in eleven dimensions

• Irreducible representations of SO(1,9) and the irreducible basis of the D=10 superspace

The following sections are included:

• Introduction

• Super Yang-Mills theories in D=4

• The Action Principle for N=1, D=10 super Yang-Mills theory

The following sections are included:

• HISTORICAL REMARKS AND REFERENCES FOR PART TWO

• PART TWO REFERENCES

The following sections are included:

• CONTENTS

Supergravity theories are the “gauge theories” of the super Poincaré algebras, discussed in Part Two…

The following sections are included:

• Local supersymmetry and gravity

• Space-time Lagrangian of D=4, N=1 supergravity

• The equations of motion of D=4, N=1 supergravity

• Supersymmetry transformations and action invariance

• On-shell supersymmetry invariance

• The linearized theory of supergravity

• Appendix 111.2.A. Commutator of Two Supersymmetries on the Gravitino Field

The following sections are included:

• From space-time to superspace

• Geometry of superspace

• The rheonomy principle

• An extended action principle

• D=4, N=1 supergravity and rheonomy

• Rheonomic constraints and Bianchi identities

• On-shell supersymmetry

• Action invariance and off-shell supersymmetry

• Building rules for supergravity Lagrangians

• Retrieving N=1, D=4 supergravity from the building rules

• Extension to anti de Sitter supergravity

• Building rules for supergravity theories using rheonomy and Bianchi identities

The following sections are included:

• Introduction

• Rheonomic solution of the N=2, D=4 Bianchi identities

• The Lagrangian of N=2, D=4 supergravity

The following sections are included:

• Introduction

• Identification of the supergroup and construction of its curvatures

• Construction of the Lagrangian

• Superspace equations of motion and on-shell supersymmetry

• The second order formulation and the contracted version of the theory

The following sections are included:

• Introduction

• The concept of free differential algebra

• The structure of free differential algebras and some theorems

• Gauging of the free differential algebras and the building rules revisited

• The Sohnius-West model (New minimal N=1 supergravity): the on-shell formulation

• The Sohnius-West model: off-shell extensions

• The building rules in their final form

The following sections are included:

• Introduction

• D=6 Weyl spinors and self dual tensors

• The free differential algebra of D=6 supergravity

• Construction of the model

• Non-invariance of the space-time action and how to cure it

The following sections are included:

• Introduction

• Free differential algebra of D=11 supergravity

• Extended F.D.A. and the introduction of a 6-form

• The gauging of F.D.A. revisited

• Constructing the theory from Bianchi identities

• The action of D=11 supergravity

• The completion of the action and the equations of motion

• Historical Remarks and References for Part Three

• REFERENCES

So far we have considered only pure supergravities which did not involve scalar fields: that is we have restricted our attention to the field theory of the graviton multiplet discarding, moreover, all the cases where this latter includes spin zero particles (N ≥ 4 in D=4, for instance)…

The following sections are included:

• σ-models of supergravity and complex manifolds

• Almost complex and complex structures on a 2n-dimensional manifold

• Hermitean and Kähler metrics

• The differential geometry of Kähler manifolds

The following sections are included:

• Kähler Geometry for the N=1 coupling

• Solution of the Bianchi identities and auxiliary fields

• Construction of the action: generalities

• Construction of ℒ(inv)

• Construction of Δℒ

The following sections are included:

• Killing vectors and isometries of the scalar manifold

• The vector multiplet

The following sections are included:

• Introduction

• The mass relation in the minimal coupling case

• Examples and flat potentials

The following sections are included:

• How to extend the symmetries of the non linear σ-model to the vector fields.

• The coset structure of extended supergravities in D=4

The following sections are included:

• Introductory remarks

• The N=3 vector multiplet and the G/H structure of the supergravity coupling

• SU(3,n)/SU(3)× SU(n)× U(1) formalism and the solution of Bianchi identities

• The Lagrangian

• The scalar field potential

• APPENDIX A: THE SCALAR POTENTIAL

• APPENDIX B: THE A_ΛΣ MATRIX AND THE EMBEDDING OF SU(3,N) INTO THE SYMPLECTIC GROUP

The following sections are included:

• Introduction to partial supersymmetry breaking

• Features of the N=3 theory and of its potential

• A short discussion of simple N=4 supergravity

• A short discussion of matter coupled N=4 supergravity

The following sections are included:

• Introduction

• Classification of D=4 supergravities and guide to the related literature

• The N=8 theory

• Results for matter coupling in D=4 supergravities

• REFERENCES

Geometric symmetries, such as space-time translations or rotations, are usually distinguished from internal symmetries, e.g. the local U(l) of electromagnetism…

The following sections are included:

• Spontaneous compactification of D=5 pure gravity

• Symmetries in D=4

• A preliminary example: the spectrum of M₄ × S¹ Maxwell theory

• The spectrum of M₄ × S¹ gravity

The following sections are included:

• H-harmonics on G/H

• Harmonic expansions in Kaluza-Klein theories

• Yang-Mills fields from M₄ × MK compactifications

The following sections are included:

• The D=4 vacuum: maximal symmetry

• AdS⁴ × M⁷ solutions (Freund-Rubin)

• Properties of the internal space M⁷ : Killing spinors and Weyl holonomy

• Osp(4/N) formulation

• Differential operators on M⁷

• APPENDIX V.4.1: SO(7) Г-MATRICES

The following sections are included:

• The linearized field equations of D=11 supergravity

• Fermion masses

• Boson masses

• Supersyrometric mass relations

• Vacuum stability

The following sections are included:

• Homogeneous 7-manifolds

• The spaces SU(3) × SU(2) × U(1)/SU(2) × U(1) × U(1)

• The other D = 7 Einstein spaces G/H

The following sections are included:

• How to compute on the G/H harmonics

• The spectrum of the round S⁷ : harmonic analysis

• The spectrum of the round seven-sphere: Osp(4/8) analysis
  • Lie superalgebras in terms of boson and fermion oscillators
  • UIR’s of noncompact supergroups with a Jordan structure
  • Oscillator representation of the noncompact Osp(4/8, R) superalgebra
  • Irreducible representations of U(2/4)
  • The construction of UIR’s of Osp(4/8, IR)
  • Decomposition of Osp(4/8) UIR’s into Sp(4)× SO(8) UIR’s
  • The singleton representation
  • he massless representation
  • The massive representations

The following sections are included:

• The M^pqr spaces

• Harmonics on the M^pqrspaces

• The spectrum of the SU(3) × SU(2) × U(1) irreps in the M^pqr spinor expansion

• Conjugation in the longitudinal spectrum

• Calculation of the longitudinal mass eigenvalues

The following sections are included:

• Introduction

• Nonvanishing internal photon (Englert-type solutions)

• Symmetries of Englert-type solutions

• Stretched and warped solutions

In this chapter we discuss, without explicit details, the problem of extracting a D=4 “low energy” supergravity from a nontrivial M⁷ compactification of D=11 S.G. A full discussion would necessarily be very technical, and there are still aspects of the D=4 embedding into D=11 which are not fully understood…

In this chapter we address the problem of obtaining chiral fermions in Kaluza-Klein theories. Most of the discussion is based on Ref. [30]…

The following sections are included:

• BIBLIOGRAPHICAL NOTE

The following sections are included:

• CONTENTS

The topics treated in the conclusive part of this book are, at the time of writing, the subject of current research. Hence a systematic illustration of all the results is premature: indeed the field is in rapid evolution and our understanding of string theory deepens and broadens by the day…

The following sections are included:

• Introduction

• Definition of a Riemann surface; metrics, complex structures and moduli space

• The simply connected Riemann surfaces and the uniformization theorem

• Deformation of the metric, quadratic differentials and the complex structure of Teichmüller space.

• Homology bases, abelian differentials and the period matrix

• Dehn twists, the mapping class group and its homomorphism onto Sp(2g,Z)

• The group of divisors and the Riemann-Roch theorem

• The Jacobian variety: Riemann theta functions and spin structures

• References quoted in Chapter VI. 2

The following sections are included:

• Introduction

• N = l, D = 2 conformal supergravity and the heterotic superspace geometry

• Classical superconformal theories and the WZW-action

• Heterotic σ-model on a general target space and the choice of M_target leading to a classical superconformal theory.

• Canonical quantization of the heterotic WZW-model and the superconformal algebra

• APPENDIX TO CHAPTER VI .3 Rules for the Wick rotation of spinors

• References for Chapter VI.3

The following sections are included:

• Introduction

• BRST quantization. Abstract properties of Q

• Construction of Q

• The BRST invariant hamiltonian and the Fradkin-Vilkovski theorem

• BRST quantization of string theories
  • The bosonic string
  • The BRST charge in complex variables
  • The NSR superstring

• References quoted in Chapter VI.4

The following sections are included:

• Introduction

• The BRST charge: cancellation of the conformal anomaly, boundary conditions and intercepts

• Twisted Kač-Moody Algebras and Massless Target Fermions

• References for Chapter VI.5

The following sections are included:

• Introduction

• The cosmological constant, the partition function, and the Polyakov path integral

• Operational evaluation of the bosonic string partition function

• The Polyakov integration measure for the bosonic string

• Functional evaluation of the bosonic string partition function in the case of the torus

• Functional determinants of the Laplacian and of the Dirac operator on the torus

• The gravitino ghost

• References for Chapter VI.6

• APPENDIX VI.6.A Detailed Treatment of the Conformal Killing Vectors

The following sections are included:

• Introduction

• Modular invariance and GNO fermionization

• Modular invariance and spin structures

• An example in D = 10: the SO(32) superstring

• A second example in D = 10: the E₈ ⊗ E₈^' and SO(16) ⊗ SO(16) heterotic strings

• Examples in D = 4

• References for Chapter VI.7

The following sections are included:

• Introduction

• Quantum Conformal Field Theories and Emission Vertices

• Bosonization, vertex operators and spin fields in the matter sector

• b-c Systems, Superghost Bosonization and the Background Charge

• The Covariant Lattice for D=10 Superstrings

• Conjugacy classes and GSO projectors: The SO(32) example in D=10

• Massless emission vertices and the effective theory of D=10 superstrings

The following sections are included:

• Introduction

• The algebraic basis of D=10, N=l matter-coupled supergravity

• The general solution of the D=10 super Poincaré Bianchi identities

• The H-Bianchi identity in the (0,4)- and (1,3)-sectors: determination of the H-parametrization

• The (2,2)- and (3,1)-sectors of the - Bianchi identity and the equations of motion

• The Lagrangian of N=l, D=10 matter coupled supergravity at γ=0

• Retrieving the superspace constraints from the κ symmetry of the Green-Schwarz string formulation

• Bianchi identities and off-shell formulations of N=l, D=4 supergravity revisited

• Chiral multiplets, the linear multiplet and the geometrical interpretation of R-symmetry

• D=4 Chern-Simons cohomology and the linear multiplet

• References for Chapter VI.9

The following sections are included:

• Introduction

• Type II superstrings on SU(2)³ groupfolds

• Construction of modular invariants and GSO projectors for the Type II superstring

• SU(2)³ groupfolds and superconformal field theories

• The h-map

• Emission vertices of the massless multiplets in an N=l heterotic model based on a (2,2)_9,9 internal theory

• Emission vertices of the massless multiplets in N=2 heterotic models based on a (4,4)_6,6 ⊕ (2,2)_3,3 internal theory

• Embedding of a (2,2)_9,9 into the direct sum (4,4)_6,6 ⊕ (2,2)_3,3

• Classification of the SU(2)³ groupfold realizations of the internal conformal field theory

• Details of the SU(2)³ groupfold construction with emphasis on bosonization

• Appendix VI.10.A. A bosonizable (2,2) vacuum of type A: A1(1,2,3,4)ε₁₃

• Appendix VI.10.B. A Bosonizable (2,2) Vacuum of Type B: B5(1,2,3,4)ε₂₃

• Appendix VI.10.C. An LRP (2,2) Vacuum of Type B: B26(1,2,3)

• Bibliographical Note

• Historical Remarks and References

• Index