Introductory to Supersymmetry and Supergravity

The following sections are included:

• 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. This included almost all of rigid supersymmetry and N = 1 supergravity. Excluded were the extended supergravity theories, superstring theories, Kaluza-Klein dimensional reduction and an extensive discussion of the phenomenological implications of supersymmetry. The reader is referred to Appendix B for some reviews on these important subjects.

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 Poincaré 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. Coleman and Mandula⁴ showed on very general assumptions that any Lie group which contained the Poincaré group P, whose generators P_a and J_ab satisfy the relations

Although the supersymmetry algebra was presented in Chapter 2 from the viewpoint of allowing the Poincaré 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. That is, in the classical limit a boson A and fermion χ_α obey the relations

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. The fields λ_α and A_a are subject to their free field equations, i.e.

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.¹⁰…

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.¹² This method consists of finding a representation of a subgroup of the Poincaré group and boosting it up to a representation of the full group. In practice, one adopts the following recipe: We choose a given momentum q^μ which satisfies P_μ² = 0 or P_μ² = - m² depending which case we are considering. We find the subgroup H which leaves q^μ intact and find a representation of H on the | q^μ states. We then induce this representation to the whole of the Poincaré group in the usual way. In this construction there is a one-to-one correspondence between points of P/H and four-momentum which satisfies P_μ² = 0 or P_μ² = - m². One can show that the result is inde pendent of the choice of momentum q^μ one starts with…

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^a and ψ_μ^α terms in w_μ^ab…

Given any symmetry group there are rules for taking any two representations and obtaining a third. In conjunction with the rules for building invariants these rules enable the systematic construction of actions. This is true not only for internal groups, but also for space-time groups such as the Poincaré group or the conformal group. For the case of the Poincare group, we work with four vectors and higher rank tensors. The rules for combining tensors to obtain tensors of a higher rank are obvious. Invariants are obtained by contraction with the Minkowski metric η_μv

A similar analysis can be given for the supersymmetry group, and comes under the name of tensor calculus. In the rigid case it was worked out by Wess and Zumino.²⁰ The rules of tensor calculus are equivalent to the superspace manipulations of Chapter 14. Although the superspace formulation is in some ways more elegant and certainly more useful at the quantum level, it is the tensor calculus which is most useful for constructing classical actions, in particular, the couplings of supergravity to matter. The reason for this is that tensor calculus works with component fields grouping them into sets (supermultiplets) which have well-defined transformation rules under supersymmetry. As such the effect of any manoeuvre on the supermultiplets can easily be evaluated in terms of expressions involving component fields.

In this chapter we wish to construct theories of rigid supersymmetry in x-space, their superspace formulations being given later. The possible on-shell states of supersymmetric massless theories with spins one and less are given by the irreducible representations of supersymmetry discussed in Chapter 8 and are listed in the table below…

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.³¹…

In this chapter, we wish to find a method of formulating supersymmetric theories, that keeps the supersymmetry manifest at every step of the calculation. This is particularly useful for the quantization of supersymmetric theories.

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

Perhaps the major use of the superfield description of supersymmetric theories is in the calculation of quantum effects. Superfields allow one to calculate the quantum behavior of supersymmetric theories, while keeping the supersymmetry manifest. The advantage this presents is analogous to calculating with usual theories in a Lorentz covariant formulation. In fact, super-Feynman rule calculations can, in some cases, be enormously easy to carry out; even easier than the corresponding x-space theory with only one of the x-space component fields…

In Chapter 17 we noted how the interacting Wess-Zumino model had even fewer infinite renormalization factors than one might naively expect by demanding that the renormalization procedure preserve supersymmetry. In fact, it only has one infinite renormalization factor as does the N = 1 Yang-Mills theory (in the background-field method or in a preferred gauge). As one studies theories with more supersymmetries one might expect an even more remarkable ultraviolet behaviour…

In this chapter, we wish to explain some of the general features of the present realistic models of sypersymmetry. In particular, we would like to explain why supersymmetry is broken at some intermediate scale (i.e. 10¹⁰GeV) and why it was so desirable to include supergravity. We do this by discussing possible ways of breaking supersymmetry; it will emerge that there are tight theoretical constraints on how supersymmetry can be broken and on the resulting mass spectra. The need to find a model, in which the mechanism of supersymmetry breaking leads to sufficiently large masses for the unobserved scalar superpartners of the quarks and leptons, leads to the form of the present realistic models…

The following sections are included:

• General Considerations

• Currents in the Wess-Zumino Model¹⁶⁵

• Currents in N = 1 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

Before beginning to covariantly quantize an extended object, namely the string, it will be instructive to consider the corresponding path from the classical to the second quantized point particle…

The following sections are included:

• Appendix A An Explanation of our Choice of Conventions

• Appendix B A List of Reviews and Books

• References