T-duality, D-branes and Brane Configurations

This section has the purpose of introducing basic ideas about T-duality in closed and open string theory. The open string case leads in a natural way to the definition of D-branes (for reviews of D-branes see [4,10,11]). These objects are of extreme importance since they are precisely the solitonic degrees of freedom which realize the strong-weak coupling duality in superstring theory. This duality is also known as string S-duality. T and S dualities relate the five perturbative superstring theories discussed previously and their compactifications in diverse dimensions. Moreover, the strong coupling limit of HE and Type II string theory (and their compactifications) suggest that there is an eleven-dimensional theory which has the eleven-dimensional supergravity as low energy limit. This prospect of theory is widely known as M-theory. The name come from the words: mystery, magic, mother, etc. Compactifications to diverse lower dimensions than ten gives more evidence of the existence of this theory. The fundamental degrees of freedom of this unified theory are unknown, but macroscopically they include membranes and fivebranes. `Matrix Theory' is an attempt to give the dof's of M-theory. The proposal is that these degrees of freedom are the D0-branes. The worldvolume effective theory of a gas of $N$ D0-branes is a SU$(N)$ quantum mechanics. Large $N$-limit reproduces the description of membranes and fivebranes and some other results of eleven dimensional supergravity (for some reviews the reader can consult [12,13]). D-branes also, are very important tools to study the strong coupling of supersymmetric theories in various dimensions. Different properties as chirality, dualities etc. are encoded in the engineering of brane configurations. The moduli space of these susy gauge theories is described by the Higgs and the Coulomb branches of the corresponding brane configuration. Many field theory results are understanding in terms of a geometrical language and many generalizations have been established motivated by the brane engineering (more about this topic can be found in Ref. [14]). Finally, the presence of branes leads to modify the prescription of Calabi-Yau or orbifold compactifications and new non-perturbative are possible. In these sections we will discuss some of these interesting topics. truecm