Introduction to Supersymmetry and Supergravity

The following sections are included:

• Preface to the Second Edition

• Preface

• Contents

This book has grown out of a number of lectures given at various summer schools and is intended to be an introduction to Supersymmetry and Supergravity. It is pedagogical in the sense that complete proofs are almost always given. After more than ten years of intensive work on supersymmetric theories, it is impossible to cover the vast literature in a single volume. Rather than discuss many topics in an incomplete way, we have decided to restrict our attention to those areas which are essential for a beginner to learn, are important for the development of the subject as a whole and can be covered in a reasonably short space in a pedagogical manner…

In the 1960's, with the growing awareness of the significance of internal symmetries such as SU(2) and larger groups, physicists attempted to find a symmetry which would combine in a non-trivial way the space-time Poincare group with an internal symmetry group. After much effort it was shown that such an attempt was impossible within the context of a Lie group…

Although the supersymmetry algebra was presented in Chapter 2 from the viewpoint of allowing the Poincare group and an internal symmetry group to coexist in a non-trivial manner, it can also be obtained, with hindsight, from at least two other approaches…

The supercharge Q_αⁱ is in the spin-½ representation of the Lorentz group and so its action on a state of spin j will result in a state of spin j ± ½. Hence, supersymmetry is a symmetry which mixes particles of different spin, that is mixes fermions and bosons. The Z₂ graded structure of the supersymmetry is now seen to be a necessary consequence of the fact that bosons and fermions obey Bose-Einstein and Fermi-Dirac statistics respectively…

The first four-dimensional model in which supersymmetry was linearly realized was found by Wess and Zumino,³ by studying two-dimensional dual models.⁷ In this chapter we rediscover supersymmetry along the lines given in Chapter 4 and discuss the Wess-Zumino model which is the simplest model of N = 1 supersymmetry…

For simplicity we will first consider an Abelian gauge group. The construction of this model follows the same pattern as the Wess-Zumino model except for the added complication of gauge invariance. The first step is to find the irreducible representation which is carried by the on-shell states which consist of one Majorana spinor, λ_α and one massless vector A_a…

Any theory whose nonlinear form is determined by a gauge principle can be constructed by a Noether procedure.⁹ Because of the importance of the Noether technique in constructing theories of supergravity we will take this opportunity to illustrate the technique within the framework of the simpler theory of supersymmetric Yang-Mills theory…

In this chapter we wish to find the irreducible representations of supersymmetry, or, put another way, we want to know what is the possible particle content of supersymmetric theories. As is well known the irreducible representations of the Poincaré group are found by the Wigner method of induced representations…

We now wish to construct supergravity along similar lines to the method used to find the Wess-Zumino model and N = 1 super QED. In this section we will start with the on-shell states and construct the linearized theory, without and then with auxiliary fields. Because of the complexity of supergravity it is particularly useful to examine the much simpler linearized theory which displays many of the features of the full nonlinear theory…

At first sight the presence of terms quartic in ψ_μα in the supergravity action would seem to indicate that it may be very lengthy to prove that the action of Eq. (9.27) is actually invariant under the transformations of Eq. (9.24). These difficult terms result from expressing w_μ^ab in terms of e_μ^a and ψ_μ^α. A rather straightforward proof of invariance^18,19 can be achieved by judicious handling of the e_μ^α and ψ_μ^α terms in w_μ^ab…

The following sections are included:

• Supermultiplets

• Combination of Supermultiplets

• Action Formulas

The following sections are included:

• N = 2 Yang-Mills

• N = 2 Matter

• The General N = 2 Rigid Theory

• The N = 4 Yang-Mills Theory

The most popular method of constructing realistic models of supersymmetry is within the context of supergravity (see Chapter 19). In order to build these models it is important to know the most general coupling of supergravity to matter. In certain quarters there is an impression that this coupling was handed down on the Tables of Moses. In fact, it was constructed using the local tensor calculus, which is the generalization to supergravity of the rigid tensor calculus of Chapter 11. Various matter coupling had previously been constructed using the Noether method…

The following sections are included:

• An Elementary Account of N = 1 Superspace

• N = 1 Superspace

• N = 2 Superspace

The following sections are included:

• N = 1 Superspace Theories. The Wess-Zumino Model

• N = 1 Yang-Mills Theory

• A Geometrical Approach to N = 1 Supersymmetric Yang-Mills Theory

• N = 2 Superspace Theories

The following sections are included:

• Geometry

• The Superspace Constraints

• Analysis of the Superspace Constraints

• Superspace Supergravity from x-space Supergravity
  • On-Shell
  • Off-Shell

The following sections are included:

• General Formalism

• The Wess-Zumino Multiplet

• Super Yang-Mills Theory

• Applications of N = 1 Super-Feynman Rules

• Divergence in Super-Feynman Graphs

• One-loop Infinities in a General N = 1 Supersymmetric Theory

• The Background-Field Method

• The Superspace Background-Field Method

The following sections are included:

• The Anomalies Argument

• The Non-Renormalization Argument

• Finite N = 2 Supersymmetric Rigid Theories

• Explicit Breaking and Finiteness

The following sections are included:

• Tree-level Breaking of Supersymmetry

• Quantum Breaking of Supersymmetry

• The Gauge Hierarchy Problem

• Comments on the Construction of Realistic Models

The following sections are included:

• General Considerations

• Currents in the Wess-Zumino Model

• Currents in N = 2 Super Yang-Mills Theory

• Quantum Generated Anomalies

• Currents and Supergravity Formulations

The following sections are included:

• 2-Dimensional Models of Rigid Supersymmetry

• Coupling of 2-Dimensional Matter to Supergravity

The following sections are included:

• Conventions in Two-Dimensional Minkowski and Euclidean Space
  • Minkowski Spinors
  • Euclidean Spinors

• Superalgebras in Two-Dimensions

• Irreducible Representations of Two-Dimensional Supersymmetry

The following sections are included:

• Minkowski Superspaces
  • (1, 0) Superspace
  • (1, 1) Superspace
  • (2, 0) Superspace
  • (2, 2) Superspace

• Euclidean Superspace
  • (1, 1) Superspace
  • (2, 2) Superspace

The following sections are included:

• Geometrical Framework

• (1, 0) Supergravity
  • The constraints
  • Solving the Bianchi identities
  • Recovery of the x-space fields
  • Local Matter Multiplets
  • The Heterotic string in superspace
  • Super-Weyl transformations
  • Solution of the constraints

• (1, 1) Supergravity

The following sections are included:

• The Conformal Group in Arbitrary Dimensions

• The Two-Dimensional Conformal Group

• The (1, 1) Superconformal Group

• The (2, 2) Superconformal Group

The following sections are included:

• Two and Three Point Green Functions

• Chiral Correlators in (2, 2) Super Conformal Models

• Super Operator Product Expansions

The following sections are included:

• The Point Particle

• The Bosonic String

• Oscillator Formalism

• The Gauge Covariant Theory at Low Levels

• The Finite Set

• The Infinite Set

• The Master Set

• The On-shell Spectrum of the Master Set

The following sections are included:

• Appendix A An Explanation of our Choice of Conventions

• Appendix B A List of Reviews and Books

• Appendix C Problems

• References

• Subject Index