String Dualities and M-Theory
The five superstring theories which we have encountered so far will be argued to be different perturbative limits of one underlying theory, connected via a web of perturbative and non-perturbative dualities. We will discuss the concept of dualities and the role of BPS states in establishing non-perturbative dualities. We construct brane solutions of supergravity, discuss their BPS properties and identify those carrying R-R charge as D-branes. We discuss the place of eleven-dimensional supergravity in this duality web and the self-consistent evidence for the existence of a so far unknown eleven-dimensional quantum gravity theory, called M-theory. The known facts include that its low-energy effective field theory is given by eleven-dimensional supergravity and that it contains membranes and five-branes, which upon compactification are related to the branes of string theory. Two examples of the power of dualities will be discussed in the final two sections: F-theory, an interesting class of non-perturbative compactifications of type IIB string theory and the AdS-CFT correspondence. They derive their appeal and usefulness from the fact that non-perturbative effects can be related to simple geometric structures.