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.