The magnets in a circular accelerator are placed in a sequence, or lattice, that is intended to return an ideal test particle, one turn later, to the same location from where it was launched. Usually, this ideal location – the design orbit – is at the centre of the vacuum beampipe, as illustrated in Figure 2.1. It can be shown mathematically that, with static fields (dB/dt = dE/dt = 0), there is one orbit – the closed orbit – that exactly repeats itself, turn after turn. In practice, the design orbit and the closed orbit are not the same, because of the inevitable presence of errors of various sorts, such as magnet misalignments.

Errors, and their correction, are discussed in Chapters 6 and 8. For now, it is assumed that the closed orbit and the design orbit are identical – there are no errors. Even so, it is necessary to consider non-ideal test particles that deviate at least a little from the design orbit. Particles circulate the accelerator in bunches, perhaps 109 per bunch. Their stability must be guaranteed as they oscillate horizontally and vertically, relative to an ideal particle at the centre of a bunch. Figure 2.2 shows the right-handed co-ordinate system that is commonly used to describe such betatron oscillations, with x and y denoting the horizontal and vertical displacements of the test particles, respectively.

Stable Oscillations

The horizontal displacement of a stable particle at a reference point on turn n is written

where a and ϕ0 are the amplitude and the initial phase of the betatron oscillation. The fractional part of the horizontal betatron tune Qx is typically not small in practice – it is 0.155 in the example shown in Figure 2.3 – and the integer part of Qx is usually much greater than 1. An arbitrary integer multiple of 2π can be added to the argument of the sine in Equation 2.1, and so the integer component does not matter when only a single reference point is being considered. This accounting must be done correctly, however, when multiple reference points are used – for example, when considering the linear motion from one reference point to another around a storage ring.

In the first part of this book we have discussed (super-)string theory in flat 26(10)- dimensional backgrounds. It is clear that such string theories are phenomenologically not relevant. Our main task in the second part is to understand some of the technology that is used to build string theories that are relevant, in particular for 4-dimensional physics. The basic idea is to study the motion of superstrings on backgrounds in which some of the dimensions are curled up. This process is known as compactification. As we shall discuss, describing strings in curved backgrounds is a highly non-trivial task. We shall sketch a few of the basic technologies in the next three chapters. This will be followed by two more blocks of chapters, one on Calabi-Yau compactifications, the other on string dualities, with special emphasis on the AdS/CFT correspondence and its application to gauge theory.

The first part of this book deals with strings that move in flat Minkowski space. While these string theories do not describe nature, their study shall allow us to illustrate many key concepts and constructions. In particular, we will be able to explore the precise relation between string theory and field theory on the world-sheet, the quantization of both closed and open strings, and the construction of consistent superstring theories. Along the way we will learn how to perform basic computations of scattering amplitudes and become quite fluent in switching between world sheet and space-time interpretation of these calculations.

The aim of this final chapter is to give an idea of the powerful applications of gauge/string dualities at the example of Maldacena's original correspondence between the maximally supersymmetric YANG-MILLS (SYM) theory in four dimensions and the dual string theory on AdS5 × S5. We will begin with a brief review of N=4 SYM theory, with special emphasis on its superconformal symmetry. The latter contains a generator of dilations that has been studied extensively after a remarkable observation by Minahan and Zarembo that, in leading order of perturbation theory, the dilation operator of N=4 SYM theory is the HAMILTONIAN of a Heisenberg spin chain [44]. We will then turn to the dual string theory and discuss some classical solutions, including that of a folded spinning string found originally by Gubser, Klebanov, and Polyakov in [34]. This solution plays a central role in modern applications of string theory to the multi-color limit of N=4 SYM theory, which we can only just touch upon in a short outlook.

Our discussion will finally bring us back to the very beginning of these chapters and to the origin of string theory as a candidate theory for hadronic physics. It shall discuss how dualities between gauge and string theory manage to circumvent the difficulties physicists faced in the early 1970s in applying string theory to strong interactions.

N=4 Super Yang-Mills Theory

We have introduced the field content and action of N=4 SYM theory at the very end of Chapter 11, as a dimensional reduction of 10-dimensional N=1 SYM theory. In addition, we have stated in the previous chapter that N=4 SYM theory gives rise to a conformal quantum field theory. Now we want to explore the model and its conformal symmetry in a bit more detail. Conformal symmetry implies scale invariance, and hence it is a rather unusual property for a 4-dimensional gauge theory. The examples of gauge theories we are usually experiencing in nature, such as, e.g., Quantum Chromodynamics (QCD), are not scale invariant. In fact, at low energies (long distances), QCD is a theory of colorless hadrons, while at very high energies (short distances) its fundamental degrees of freedom are quarks and gluons.