It is well known that supersymmetric theories may have Bogomol'nyi–Prasad–Sommerfield (BPS) sectors in which some data can be computed at strong coupling even when the full theory is not solvable. Historically, this is how the first exact results on particle spectra were obtained [1]. Seiberg–Witten's breakthrough results [2, 3] in the mid 1990s gave an additional motivation to the studies of the BPS sectors.

BPS solitons can emerge in those supersymmetric theories in which superalgebras are centrally extended. In many instances the corresponding central charges are seen at the classical level. In some interesting models central charges appear as quantum anomalies.

First studies of BPS solitons (sometimes referred to as critical solitons) in supersymmetric theories at weak coupling date back to the 1970s. De Vega and Schaposnik were the first to point out [4] that a model in which classical equations of motion can be reduced to first–order Bogomol'nyi–Prasad–Sommerfeld (BPS) equations [5, 6] is, in fact, a bosonic reduction of a supersymmetric theory. Already in 1977 critical soliton solutions were obtained in the superfield form in some twodimensional models [7]. In the same year miraculous cancellations occurring in calculations of quantum corrections to soliton masses were noted in [8] (see also [9]). It was observed that for BPS solitons the boson and fermion modes are degenerate and their number is balanced. It was believed (incorrectly, we hasten to add) that the soliton masses receive no quantum corrections.

Let us move towards less supersymmetric theories. In this chapter we will review non-Abelian strings in four-dimensional gauge theories with N = 1. In Chapter 6 we will deal with N = 0.

As was discussed in the Introduction to Part II, the Seiberg-Witten mechanism of confinement [2, 3] relies on a cascade gauge symmetry breaking: the non-Abelian gauge group breaks down to an Abelian subgroup at a higher scale by condensation of the adjoint scalars, and at a lower scale the Abelian subgroup breaks down to a discrete subgroup by condensation of quarks (or monopoles, depending on the type of vacuum considered). This leads to formation of the ANO flux tubes and ensures an Abelian nature of confinement of the monopoles (or quarks, respectively). The gauge group acting in the infrared, where the confinement mechanism becomes operative, is Abelian.

On the other hand, non-supersymmetric QCD-like theories as well as N = 1 SQCD have no adjoint scalars and, as a result, no cascade gauge symmetry breaking occurs. The gauge group acting in the infrared is non-Abelian. Confinement in these theories is non-Abelian. This poses a problem of understanding confinement in theories of this type. Apparently, a straightforward extrapolation of the Seiberg-Witten confinement scenario to these theories does not work.

The discovery of the non-Abelian strings [130, 131, 132, 133] suggests a novel possibility of solving this problem.

This concludes our travel diary in the land of supersymmetric solitons in gauge theories. It is time to summarize the lessons.

Advances in supersymmetric solitons, especially in non-Abelian gauge theories, that have taken place since 1996, are impressive. In the bulk of this book we thoroughly discussed many aspects of the subject at a technical level. Important and relevant technical details presented above should not overshadow the big picture, which has been in the making since 1973. Sometimes people tend to forget about this big picture which is understandable: its development is painfully slow and notoriously difficult.

Let us ask ourselves: what is the most remarkable feature of quantum chromodynamics and QCD-like theories? The fact that at the Lagrangian level one deals with quarks and gluons while experimentalists detect pions, protons, glueballs and other color singlet states – never quarks and gluons – is the single most salient feature of non-Abelian gauge theories at strong coupling. Color confinement makes colored degrees of freedom inseparable. In a bid to understand this phenomenon Nambu, 't Hooft and Mandelstam suggested in the mid 1970s (independently but practically simultaneously) a “non-Abelian dual Meissner effect.” At that time their suggestion was more of a dream than a physical scenario. According to their vision, “non-Abelian monopoles” condense in the vacuum resulting in formation of “non-Abelian chromoelectric flux tubes” between color charges, e.g. between a probe heavy quark and antiquark.

Dp-branes and boundary conditions

A Dp-brane is an extended object with p spatial dimensions. In bosonic string theory, where the number of spatial dimensions is 25, a D25-brane is a space-filling brane. The letter D in Dp-brane stands for Dirichlet. In the presence of a D-brane, the endpoints of open strings must lie on the brane. As we will see in more detail below, this requirement imposes a number of Dirichlet boundary conditions on the motion of the open string endpoints.

Not all extended objects in string theory are D-branes. Strings, for example, are 1-branes because they are extended objects with one spatial dimension, but they are not D1-branes. Branes with p spatial dimensions are generically called p-branes. A 0-brane is some kind of particle. Just as the world-line of a particle is one-dimensional, the world-volume of a p-brane is (p + 1)-dimensional. Of these p + 1 dimensions, one is the time dimension and the other p are spatial dimensions.

Introduction

In this book, the quantization of strings was carried out using light-cone coordinates and the light-cone gauge. String theory is a Lorentz invariant theory, but Lorentz symmetry is not manifest in the light-cone quantum theory. Indeed, the choice of a particular coordinate X+ for special treatment hides from plain view the Lorentz symmetry of the theory. While hidden, the Lorentz symmetry is still a symmetry of the quantum theory, as we demonstrated by the construction of the Lorentz generator M−I. This generator has the expected properties when the spacetime has the critical dimension.

Since Lorentz symmetry is of central importance, it is natural to ask if we can quantize strings preserving manifest Lorentz invariance. It is indeed possible to do so. The Lorentz covariant quantization has some advantages over the light-cone quantization. Our lightcone quantization of open strings did not apply to D0-branes because the light-cone gauge requires that at least one spatial open string coordinate has Neumann boundary conditions. Covariant quantization applies to D0-branes.

String theory is one of the most exciting fields in theoretical physics. This ambitious and speculative theory offers the potential of unifying gravity and all the other forces of nature and all forms of matter into one unified conceptual structure.

String theory has the unfortunate reputation of being impossibly difficult to understand. To some extent this is because, even to its practitioners, the theory is so new and so ill understood. However, the basic concepts of string theory are quite simple and should be accessible to students of physics with only advanced undergraduate training.

I have often been asked by students and by fellow physicists to recommend an introduction to the basics of string theory. Until now all I could do was point them either to popular science accounts or to advanced textbooks. But now I can recommend to them Barton Zwiebach's excellent book.

Zwiebach is an accomplished string theorist, who has made many important contributions to the theory, especially to the development of string field theory. In this book he presents a remarkably comprehensive description of string theory that starts at the beginning, assumes only minimal knowledge of advanced physics, and proceeds to the current frontiers of physics. Already tested in the form of a very successful undergraduate course at MIT, Zwiebach's exposition proves that string theory can be understood and appreciated by a wide audience.

I strongly recommend this book to anyone who wants to learn the basics of string theory.

Light-cone Hamiltonian and commutators

We are at long last in a position to quantize the relativistic string. We have acquired considerable intuition for the dynamics of classical relativistic strings, and we have examined in detail how to quantize the simpler, but still nontrivial, relativistic point particle. Moreover, having taken a brief look into the basics of scalar, electromagnetic, and gravitational quantum fields in the light-cone gauge, we will be able to appreciate the implications of quantum open string theory. In this chapter we will deal with open strings. We will assume throughout the presence of a space-filling D-brane. In the next chapter we will quantize the closed string.

Maxwell fields coupling to open strings

Among the quantum states of open strings attached to a D-brane we found photon states with polarizations and momentum along the D-brane directions. We thus deduced that a Maxwell field lives on the world-volume of a D-brane. The existence of this Maxwell field was in fact necessary to preserve the gauge invariance of the term that couples the Kalb–Ramond field to the string in the presence of a D-brane. We also learned that the endpoints of open strings carry Maxwell charge.

Since any D-brane has a Maxwell field, it is physically reasonable to expect that background electromagnetic fields can exist: there may be electric or magnetic fields that permeate the D-brane.

Area functional for spatial surfaces

The action for a relativistic string must be a functional of the string trajectory. Just as a particle traces out a line in spacetime, a string traces out a surface. The line traced out by the particle in spacetime is called the world-line. The two-dimensional surface traced out by a string in spacetime will be called the world-sheet. A closed string, for example, will trace out a tube, while an open string will trace out a strip. These two-dimensional world-sheets are shown in the spacetime diagram of Figure 6.1. The lines of constant x0 in these surfaces are the strings. These are the objects an observer sees at the fixed time x0. They are open curves for the surface describing the open string evolution (left), and they are closed curves for the surface describing the closed string evolution (right).

Introduction

String theory was discovered in the attempts to understand the dynamics of strongly interacting hadrons. It had been noted that the plot of the angular momentum J of hadronic excitations against their energy-squared falls roughly into lines J = α′E2 called Regge trajectories. String theory seemed to be a reasonable candidate for a theory of strong interactions because this relationship between J and E2 emerges naturally from a rotating classical open string, as we discussed in Section 8.6.