Classical Strings and Membranes in the AdS/CFT Correspondence

In its strongest version, the AdS/CFT conjecture states that N = 4, su (Nc) super Yang-Mills (SYM) theory is equal to type IIB string theory on AdS5 × S. It is by far the most important equation of contemporary theoretical physics, a sort of a "harmonic oscillator" for both the quantum theory of gravity and gauge theories. It goes without saying that it is imperative to fully understand its limits of validity and thoroughly investigate its implications. In particular, it would be desirable to solve the theory, i.e. to be able to compute all of its observables. One of the most important observables of AdS/CFT is its spectrum. According to the AdS/CFT "dictionary", the spectrum of the theory comprises the energies of its string states, each of which must be equal to the scaling dimensions of its dual gauge theory operator. The full spectral problem of AdS/CFT is solved by integrability, in the sense that integrability provides the full set of algebraic equations that determine it. Integrability methods are however severely limited in the regime of long, strongly coupled operators, such as those that are dual to the Gubser-Klebanov-Polyakov (GKP) strings, giant magnons and single spike strings. In this thesis we study classical strings and branes in the context of the AdS/CFT correspondence. Our goal is twofold: (1) develop methods for computing the AdS5/CFT4 spectrum in the case of long, strongly coupled operators, by using classical strings and (2) understand the role of classical membranes in AdS/CFT by investigating their stringy limits. With regard to the first objective, we compute the classical spectra of long rotating GKP strings, giant magnons and single spikes. The conserved linear and angular momenta of these string configurations, that live either in AdS3 or R×S, are known in parametric form in terms of the strings’ linear and angular velocities. We eliminate the linear and angular velocities from the expressions that give the energy of the strings, in favor of the strings’ conserved charges of linear and angular momenta. This way, we find all the leading, subleading and next-to-next-to-leading terms in the dispersion relations of the aforementioned string configurations. Our results are expressed in closed forms with Lambert’s W-function. For the second objective we introduce and study "stringy membranes", a new class of membranes that live in AdS4/7 × S or AdS4 × S/Zk and have the same equations of motion, constraints and conserved charges with strings that live in an appropriate subset of AdS5. Stringy membranes can be constructed whenever the target spacetime contains a compact submanifold, by identifying one of the submanifold’s compact coordinates with one of the membrane worldvolume coordinates. For the stringy membranes that reproduce the pulsating and rotating GKP strings in AdS, we find that the spectrum of their transverse quadratic fluctuations displays a multiple band/gap structure governed by the Lamé equation. Conversely, string excitations are represented by a single-band/single-gap Lamé pattern. These findings confirm the picture that we have of membranes as collective excitations of some stringy counterparts.