Physical phenomena at energies accessible in today's accelerator laboratories are accurately described by the standard model, the renormalizable theory of quarks, leptons, and gauge bosons, governed by the gauge group SU(3) × SU(2) × U(1), described in Sections 18.7 and 21.3. The standard model is today usually understood as a low-energy approximation to some as-yet-unknown fundamental theory in which gravitation appears unified with the strong and electroweak forces at an energy somewhere in the range of 1016 to 1018 GeV. This raises the hierarchy problem: what accounts for the enormous ratio of this fundamental energy scale and the energy scale ≈ 300 GeV that characterizes the standard model?

The strongest theoretical motivation for supersymmetry is that it offers a hope of solving the hierarchy problem. Quarks, leptons, and gauge bosons are required by the SU(3) × SU(2) × U(1) gauge symmetry to appear with zero masses in the Lagrangian of the standard model, so that the physical masses of these particles are proportional to the electroweak breaking scale, which in turn is proportional to the mass of the scalar fields responsible for the electroweak symmetry breakdown. The crux of the hierarchy problem1a is that the scalar fields, unlike the fermion and gauge boson fields, are not protected from acquiring large bare masses by any symmetry of the standard model, so it is difficult to see why their masses, and hence all other masses, are not in the neighborhood of 1016 to 1018 GeV.

Now we know the structure of the most general supersymmetry algebras, and we have seen how to work out the implications of this symmetry for the particle spectrum. In order to learn what supersymmetry has to say about particle interactions, we need to see how to construct supersymmetric field theories.

Originally the construction of field supermultiplets was done directly, by a repeated use of the Jacobi identities, much as in the construction of supermultiplets of one-particle states in Sections 25.4 and 25.5. Section 26.1 presents one example of this technique, used here to construct supermultiplets containing only scalar and Dirac fields. Fortunately there is an easier technique, invented by Salam and Strathdee, in which supermultiplets of fields are gathered into ‘superfields,’ which depend on fermionic coordinates as well as on the usual four coordinates of spacetime. Superfields are introduced in Section 26.2, and used to construct supersymmetric field theories and to study some of their consequences in Sections 26.3-26.8. This chapter will be concerned only with N = 1 supersymmetry, where the superfield formalism has been chiefly useful. At the end of the next chapter we will construct theories with N-extended supersymmetry by imposing the U(N) /∧-symmetry on theories of N = 1 superfields.

Direct Construction of Field Supermultiplets

To illustrate the direct construction of a field supermultiplet, we will consider fields that can destroy the particles belonging to the simplest supermultiplet of arbitrary mass discussed in Section 25.5: two spinless particles and one particle of spin 1/2.

We have encountered a number of infinite-dimensional symmetry algebras on the world-sheet: conformal, superconformal, and current. While we have used these symmetries as needed to obtain specific physical results, in the present chapter we would like to take maximum advantage of them in determining the form of the world-sheet theory. An obvious goal, not yet reached, would be to construct the general conformal or superconformal field theory, corresponding to the general classical string background.

This subject is no longer as central as it once appeared to be, as spacetime rather than world-sheet symmetries have been the principal tools in recent times. However, it is a subject of some beauty in its own right, with various applications to string compactification and also to other areas of physics.

We first discuss the representations of the conformal algebra, and the constraints imposed by conformal invariance on correlation functions. We then study some examples, such as the minimal models, Sugawara and coset theories, where the symmetries do in fact determine the theory completely. We briefly summarize the representation theory of the N = 1 superconformal algebra. We then discuss a framework, rational conformal field theory, which incorporates all these CFTs. To conclude this chapter we present some important results about the relation between conformal field theories and nearby two-dimensional field theories that are not conformally invariant, and the application of CFT in statistical mechanics.

In the final four chapters we would like to see how compactification of string theory connects with previous ideas for unifying the Standard Model. Our primary focus is the weakly coupled E8 × E8 heterotic string, whose compactification leads most directly to physics resembling the Standard Model. At various points we consider other string theories and the effects of strong coupling. In addition, compactified string theories have interesting nonperturbative dynamics, beyond that which we have seen in ten dimensions. In the final chapter we discuss some of the most interesting phenomena.

The two main issues are specific constructions of four-dimensional string theories and general results derived from world-sheet and spacetime symmetries. Our approach to the constructions will generally be to present only the simplest examples of each type, in order to illustrate the characteristic physics of compactified string theories. On the other hand, we have collected as many of the general results as possible.

String compactifications fall into two general categories. The first are based on free world-sheet CFTs, or on CFTs like the minimal models that are solvable though not free. For these one can generally determine the exact tree-level spectrum and interactions. The second category is compactification in the geometric sense, taking the string to propagate on a smooth spacetime manifold some of whose dimensions are compact. In general one is limited to an expansion in powers of α′/R2c, with Rc being the characteristic radius of compactification. This is in addition to the usual expansion in the string coupling g.