In a standard advanced field theory course, one learns about a number of symmetries: Poincaré invariance, global continuous symmetries, discrete symmetries, gauge symmetries, approximate and exact symmetries. These latter symmetries all have the property that they commute with Lorentz transformations, and in particular they commute with rotations. So the multiplets of the symmetries always contain particles of the same spin; in particular, they always consist of either bosons or fermions.

For a long time, it was believed that these were the only allowed types of symmetry; this statement was even embodied in a theorem, known as the Coleman– Mandula theorem. However, physicists studying theories based on strings stumbled on a symmetry which related fields of different spin. Others quickly worked out simple field theories with this new symmetry: supersymmetry.

Supersymmetric field theories can be formulated in dimensions up to eleven. These higher-dimensional theories will be important when we consider string theory. In this chapter, we consider theories in four dimensions. The supersymmetry charges, because they change spin, must themselves carry spin – they are spin-1/2 operators. They transform as doublets under the Lorentz group, just like the two component spinors X and X*. (The theory of two-component spinors is reviewed in Appendix A, where our notation, which is essentially that of the text by Wess and Bagger (1992), is explained). There can be 1, 2, 4 or 8 such spinors; correspondingly, the symmetry is said to be N = 1, 2, 4 or 8 supersymmetry. Like generators of an ordinary group, the supersymmetry generators obey an algebra; unlike an ordinary bosonic group, however, the algebra involves anticommutators as well as commutators (it is said to be “graded”).

Considerations of the sort we encountered in the previous chapter have inspired two approaches to physics beyond the Standard Model: large extra dimensions (ADD) and warped spaces (Randall–Sundrum). In this chapter we will provide a brief introduction to each.

Large extra dimensions: the ADD proposal

In string theory, it is natural to imagine that the compactification scale is not too much different from the Planck scale. The size of the compact space is typically a modulus, and if it is stabilized, one might expect it be stabilized at a value not much different than one, in string (and therefore Planck) units. In terms of our general discussion of moduli stabilization, this is precisely what we would expect: once the radius becomes very large, any potential, perturbative or non-perturbative, tends to zero.

But if we are willing to discard this prejudice, an extraordinary possibility opens up. Perhaps the extra dimensions are not Planck size, but much larger, even macroscopic? Arkani-Hamed, Dimopoulos and Dvali realized that from an experimental point of view, the limits on the size of such large compact dimensions are surprisingly weak. Allowing the extra dimensions to be large totally reorients our thinking about the nature of couplings and scales in string theory (or any underlying fundamental theory). Such a viewpoint places the hierarchy problem in a whole different light, perhaps allowing entirely different solutions than technicolor or supersymmetry.

Branes are crucial to this picture. The observed gauge couplings are small, but not extremely small. But in Kaluza–Klein theory and in weakly coupled string theories, they are related to the underlying scales in a clear way.

Two of the most profound scientific discoveries of the early twentieth century were special relativity and quantum mechanics. With special (and general) relativity came the notion that physics should be local. Interactions should be carried by dynamical fields in space-time. Quantum mechanics altered the questions which physicists asked about phenomena; the rules governing microscopic (and some macroscopic) phenomena were not those of classical mechanics. When these ideas are combined, they take on their full force, in the form of quantum field theory. Particles themselves are localized, finite-energy excitations of fields. Otherwise mysterious phenomena such as the connection of spin and statistics are immediate consequences of this marriage. But quantum field theory does pose a serious challenge. The Schrödinger equation seems to single out time, making a manifestly relativistic description difficult. More serious, but closely related, the number of degrees of freedom is infinite. In the 1920s and 1930s, physicists performed conventional perturbation theory calculations in the quantum theory of electrodynamics, quantum electrodynamics or QED, and obtained expressions which were neither Lorentz invariant nor finite. Until the late 1940s, these problems stymied any quantitative progress, and there was serious doubt whether quantum field theory was a sensible framework for physics.

Despite these concerns, quantum field theory proved a valuable tool with which to consider problems of fundamental interactions. Yukawa proposed a field theory of the nuclear force, in which the basic quanta were mesons. The corresponding particle was discovered shortly after the Second World War. Fermi was aware of Yukawa's theory, and proposed that the weak interactions arose through the exchange of some massive particle – essentially the W± bosons which were finally discovered in the 1980s.

In Chapter 5, we learned a great deal about the dynamics of quantum chromodynamics. In Section 4.5, we argued that the hierarchy problem is one of the puzzles of the Standard Model. The grand unified models of the previous chapter provided a quite stark realization of the hierarchy problem. In an SU(5) grand unified model, we saw that it is necessary to carefully adjust the couplings in the Higgs potential in order that one obtain light doublets and heavy color triplet Higgs. This is already true at tree level; loop effects will correct these relations, requiring further delicate adjustments.

The first proposal to resolve this problem goes by the name “technicolor” and is the subject of this chapter. The technicolor hypothesis exploits our understanding of QCD dynamics. It elegantly explains the breaking of the electroweak symmetry. It has more difficulty accounting for the masses of the quarks and leptons, and simple versions seem incompatible with precision studies of the W and Z particles. In this chapter, we will introduce the basic features of the technicolor hypothesis. We will not attempt to review the many models that have been developed to try to address the difficulties of flavor and precision electroweak experiments. It is probably safe to say that, as of this writing, none is totally successful, nor are they terribly plausible. But it should be kept in mind that this may reflect the limitations of theorists; experiment may yet reveal that nature has chosen this path. In the second part of this book, we will argue that in string theory, ignoring phenomenological details, a technicolored solution to the hierarchy problem seems as likely as its main competitor, supersymmetry.

We don't live in a ten-dimensional world, and certainly not in a twenty-sixdimensional world without fermions. But if we don't insist on Lorentz invariance in all directions, there are other possible ways to construct consistent string theories. In this chapter we will uncover many consistent string theories in four dimensions (and in others). If anything, our problem will shortly be an embarrassment of riches: we will see that there are vast numbers of possible string constructions. The connection of these various constructions to one another is not always clear. Many of these can be obtained from one another by varying expectation values of light fields (moduli). One might imagine that others could be obtained by exciting massive fields as well. In general, though, this is not known, and, in any case, the meaning of such connections in a theory of gravity is obscure. But before exploring these deep and difficult questions, we need to acquire some experience with constructing strings in different dimensions.

Compactification in field theory: the Kaluza–Klein program

The idea that space-time might be more than four-dimensional was first put forward by Kaluza and Klein shortly after Einstein published his general theory of relativity. They argued that, in this case, five-dimensional general coordinate invariance would give rise to both four-dimensional general coordinate invariance and a U(1) gauge invariance, unifying electromagnetism and gravity. In modern language, they considered the possibility that space-time is five-dimensional, with the structure M4 × S1. This is, on first exposure, a bizarre concept, but its implications are readily understood by considering a toy model. Take a single scalar field, Φ, in five dimensions.

So far, we have focussed on rather simple models, involving toroidal compactifications and their orbifold generalizations. But while by far the simplest, these turn out to be only a tiny subset of the possible manifolds on which to compactify string theories. A particularly interesting and rich set of geometries is provided by the Calabi–Yau manifolds. These are manifolds which are Ricci flat, RMN = 0. Their interest arises in large part because these compactifications can preserve some subset of the full ten-dimensional supersymmetry. This is significant if one believes that low-energy supersymmetry has something to do with nature. It is also important at a purely theoretical level, since, as usual, supersymmetry provides a great deal of control over any analysis; at the same time, there is less supersymmetry than in the toroidal case, so a richer set of phenomena are possible.

This chapter is intended to provide an introduction to this subject. In the first section, we will provide some mathematical preliminaries. Unlike the toroidal or orbifold compactifications, it is not possible, in most instances, to provide explicit formulas for the underlying metric on the manifold and other quantities of interest. The six-dimensional Calabi–Yau spaces, for example, have no continuous isometries (symmetries), so at best one can construct the metrics by numerical methods. But it turns out to be possible to extract much important information without detailed knowledge of the metric from topological considerations. The machinery required to define these spaces and to extract at least some of this information includes algebraic geometry and cohomology theory, subjects not part of the training of most physicists.

In the Type II theory, we have seen that the left and right movers are essentially independent. At the level of the two-dimensional Lagrangian, there is a reflection symmetry between left and right movers. However, this symmetry does not hold sector by sector; it is broken by boundary conditions and projectors.

In the heterotic theory, this independence is taken further, and the degrees of freedom of the left and right movers are taken to be independent – and different. There are two convenient world sheet realizations of this theory, known as the fermionic and bosonic formulations. In both, there are eight left-moving and eight right-moving XI s, associated with ten flat coordinates in space-time. There are eight right-moving two-dimensional fermions, ψI. There is a right-moving supersymmetry, but no left-moving supersymmetry. In the fermionic formulation there are, in addition, 32 left-moving fermions which have no obvious connection with space-time, λA. In the bosonic description, there are an additional 16 left-moving bosons. In other words, there are 24 left-moving bosonic degrees of freedom. There are actually several heterotic string theories in ten dimensions. Rather than attempt a systematic construction, we will describe the two supersymmetric examples. These have gauge group O(32) and E8 × E8. The group E8, one of the exceptional groups in Cartan's classification, is not terribly familiar to most physicists. However, it is in this theory that we can most easily find solutions which resemble the Standard Model. We will introduce certain features of E8 group theory as we need them. More detail can be found in the suggested reading. In this chapter, we will work principally in the fermionic formulation.

In Chapter 18, we put forward a history of the universe. The picture is extremely simple. Its inputs are Einstein's equations and the assumptions of homogeneity and isotropy. We also used our knowledge of laws of atomic, nuclear and particle physics. We saw a number of striking confirmations of this basic picture, but there are many puzzles.

1. (1) The most fundamental problem is that we don't know the laws of physics relevant to temperatures greater than about 100 GeV. If there is only a single Higgs doublet at the weak scale, it is possible that we can extend this picture back to far earlier times. If there is, say, supersymmetry or large extra dimensions, the story could change drastically. Even if things are simple at the weak scale, we will not be able to extend the picture all the way back to t = 0. We have already seen that the classical gravity analysis breaks down.

2. (2) There are a number of features of the present picture we cannot account for within the Standard Model. Specifically, what is the dark matter? There is no candidate among the particles of the Standard Model. Is it some new kind of particle? As we will see, there are plausible candidates from the theoretical structures we have proposed, and they are the subject of intense experimental searches.

3. […]

String theory was stumbled on, more or less, by accident. In the late 1960s, string theories were first proposed as theories of the strong interactions. But, it was quickly realized that, while hadronic physics has a number of string-like features, string theories were not suitable for a detailed description. In their simplest form, string theories had massless spin-two particles and more than four dimensions of spacetime, hardly features of the strong interactions. But a small group of theorists appreciated that the presence of a spin-two particle implied that these theories were generally covariant and explored them through the 1970s and and early 1980s as possible theories of quantum gravity. Like field theories, the number of possible string theories seemed to be infinite, while, unlike field theories, there was reason to believe that these theories did not suffer from ultraviolet divergences. In the 1980s, however, studies of anomalies in higher dimensions suggested that all string theories with chiral fermions and gauge interactions suffered from quantum anomalies. But in 1984, it was shown that anomalies cancel for two choices of gauge group. It was quickly recognized that the non-anomalous string theories do come close to unifying gravity and the Standard Model of particle physics. Many questions remained. Beginning in 1995, great progress was made in understanding the deeper structure of these theories. All of the known string theories were understood to be different limits of some larger structure. As string theories still provide the only framework in which one can do systematic computations of quantum gravity effects, many workers use the term “string theory” to refer to some underlying structure which unifies quantum mechanics, gravity and gauge interactions.