The discovery that the weak forces break previously unquestioned discrete symmetries – first parity P and charge conjugation C, then CP and T – had a revolutionizing impact on our perception of nature and how we analyse the elements of its grand design. We realized that symmetries should not be taken for granted; some even began questioning that sacrosanct fruit of quantum field theory, CPT invariance. We learnt from the violation of CP symmetry – not from that of P and C separately – that left and right or positive and negative charge are dynamically distinct rather than being mere labels based on a convention; furthermore that nature distinguishes between past and future even on the microscopic level. From 1964 to 2001 CP violation had been observed only in a single system – the decays of KL mesons – as a seemingly unobtrusive phenomenon. Yet even so we had come to understand that it represents not only a profound intellectual insight, but has also many and far-reaching concrete consequences.

• The huge predominance of matter over antimatter apparently observed in our universe requires CP violation if it is to be understood as dynamically generated rather than merely reflecting the initial conditions.

• Once the dynamics are sufficiently complex to support CP violation, the latter can manifest itself in numerous different ways; we can even say the floodgates open.

• The three-family SM can implement CP violation through the KM mechanism without requiring so-far unobserved degrees of freedom. It is already highly non-trivial that it can accommodate the data on ∈K and ∈k and ∈′ within the uncertainties.

• […]

In this chapter we discuss a particular class of N = 2 supersymmetric gauge theories in which non-Abelian strings were found. One can pose the question: what is so special about these models that makes an Abelian ZN string become non-Abelian? Models we will dwell on below have both gauge and flavor symmetries broken by the condensation of scalar fields. The common feature of these models is that some global diagonal combination of color and flavor groups survive the breaking. We consider the case when this diagonal group is SU(N)C+F, where the subscript C + F means a combination of global color and flavor groups. The presence of this unbroken subgroup is responsible for the occurrence of the orientational zero modes of the string which entail its non-Abelian nature.

Clearly, the presence of supersymmetry is not important for the construction of non-Abelian strings. In particular, while here we focus on the BPS non-Abelian strings in N = 2 supersymmetric gauge theories, in Chapter 5 we review non-Abelian strings in N = 1 supersymmetric theories and in Chapter 6 in nonsupersymmetric theories.

Basic model: N = 2 SQCD

The model we will deal with derives from N = 2 SQCD with the gauge group SU(N + 1) and Nf = N flavors of the fundamental matter hypermultiplets which we will call quarks [3]. At a generic point on the Coulomb branch of this theory, the gauge group is broken down to U(1)N.

Ever since 't Hooft [124] and Mandelstam [125] put forward the hypothesis of the dual Meissner effect to explain color confinement in non-Abelian gauge theories, people were trying to find a controllable approximation in which one could reliably demonstrate the occurrence of the dual Meissner effect in these theories. A breakthrough achievement was the Seiberg–Witten solution [2] of N = 2 supersymmetric Yang–Mills theory. They found massless monopoles and, adding a small (N = 2)-breaking deformation, proved that they condense creating strings carrying a chromoelectric flux. It was a great success in qualitative understanding of color confinement.

A more careful examination shows, however, that details of the Seiberg–Witten confinement are quite different from those we expect in QCD-like theories. Indeed, a crucial aspect of Ref. [2] is that the SU(N) gauge symmetry is first broken, at a high scale, down to U(1)N-1, which is then completely broken at a much lower scale where condensation of magnetic monopoles occurs. Correspondingly, the strings in the Seiberg–Witten solution are, in fact, Abelian strings [36] of the Abrikosov–Nielsen–Olesen (ANO) type which results, in turn, in confinement whose structure does not resemble at all that of QCD. In particular, the “hadronic” spectrum is much richer than that in QCD [126, 127, 128, 35, 129].

D branes are extended objects in string theory on which strings can end [10]. Moreover, the gauge fields are the lowest excitations of open superstrings, with the endpoints attached to D branes. SU(N) gauge theories are obtained as a field-theoretic reduction of a string theory on the world volume of a stack of N D branes.

Our task is to see how the above assertions are implemented in field theory. We have already thoroughly discussed field-theoretic strings. Solitonic objects of the domain wall type were also extensively studied in supersymmetric gauge theories in 1+3 dimensions. The original impetus was provided by the Dvali-Shifman observation [11] of the critical (BPS-saturated) domain walls in N = 1 gluodynamics, with the tension scaling as. N Λ The peculiar N dependence of the tension prompted [12] a D-brane interpretation of such walls. Ideas as to how flux tubes can end on the BPS walls were analyzed [213] at the qualitative level shortly thereafter. Later on, BPS-saturated domain walls and their junctions with strings were discussed [214, 215] in a more quantitative aspect in N = 2 sigma models. Some remarkable parallels between field-theoretical critical solitons and the D-brane string theory construction were discovered.

In this and subsequent chapters we will review the parallel found between the field-theoretical BPS domain walls in gauge theories and D branes/strings. In other words, we will discuss BPS domain walls with the emphasis on localization of the gauge fields on their world volume.

One common feature of supersymmetric gauge theories is the presence of moduli spaces – manifolds on which scalar fields can develop arbitrary VEVs without violating the zero energy condition. If on these vacuum manifolds the gauge group is broken, either completely or partially, down to a discrete subgroup, these manifolds are referred to as the Higgs branches.

One may pose a question: what happens with the flux tubes and confinement in theories with the Higgs branches? The Higgs branch represents an extreme case of type-I superconductivity, with vanishing Higgs mass. One may ask oneself whether or not the ANO strings still exist in this case, and if yes, whether they provide confinement for external heavy sources.

This question was posed and studied first in [102] where the authors concluded that the vortices do not exist on the Higgs branches due to infrared problems. In Refs. [211, 212] the N = 1 SQED vortices were further analyzed. It was found that at a generic point on the Higgs branch strings are unstable. The only vacuum which supports string solutions is the base point of the Higgs branch where the strings become BPS-saturated. The so-called “vacuum selection rule” was put forward in [211, 212] to ensure this property.

On the other hand, in [103, 175] it was shown that infrared problems can be avoided provided certain infrared regularizations are applied.

In Chapter 8 we reviewed the construction of D-brane prototypes in field theory. In string theory D branes are extended objects on which fundamental strings can end. To make contact with this string/brane picture one may address a question whether or not solitonic strings can end on the domain wall which localizes gauge fields. The answer to this question is yes. Moreover, the string endpoint plays a role of a charge with respect to the gauge field localized on the wall surface. This issue was studied in [215] in the sigma-model setup and in [217] for gauge theories at strong coupling. A solution for a 1/4-BPS wall-string junction in the N = 2 supersymmetric U(1) gauge theory at weak coupling was found in [142], while [37] deals with its non-Abelian generalization. Further studies of the wall-string junctions were carried out in [172] where all 1/4-BPS solutions to Eqs. (4.5.5) were obtained, and in [218, 231] where the energy associated with the wall-string junction (boojum) was calculated, and in [232, 228] where a quantum version of the effective theory on the domain wall world volume which takes into account charged matter (strings of the bulk theory) was derived. Below we will review the wall-string junction solutions and then briefly discuss how the presence of strings in the bulk modifies the effective theory on the wall.