In this dissertation, we explore some aspects of semiclassical type IIB string theory on AdS3 x S1 and on pure AdS3 in the limit of large angular momentum S. We first focus on the integrability technique known as finite-gap formalism for strings in AdS3 x S1, leading to the definition of a hyperelliptic Riemann surface, the spectral curve, which encodes, albeit in a rather implicit fashion, the semiclassical spectrum of a very large family of string solutions. Then, we show that, in the large angular momentum limit, the spectral curve separates into two distinct surfaces, allowing the derivation of an explicit expression for the spectrum, which is correspondingly characterised by two separate branches. The latter may be interpreted in terms of two kinds of spikes appearing on the strings: 'large' spikes, yielding an infinite contribution to the energy and angular momentum of the string, and 'small' spikes, representing finite excitations over the background of the 'large' spikes. According to the AdS/CFT correspondence, strings moving in AdS3 x S1 should be dual to single trace operators in the sl(2) sector of N = 4 super Yang-Mills theory. The corresponding one-loop spectrum in perturbation theory may also be computed through integrability methods and, in the large conformal spin limit S → ∞ (equivalent to the AdS3 angular momentum in string theory) is also expressed in terms of a spectral curve and characterised in terms of the so-called holes. We show that, with the appropriate identifications and with the usual extrapolation from weak to strong 't Hooft coupling described by the cusp anomalous dimension, the large-S spectra of gauge theory and of string theory coincide. Furthermore, we explain how 'small' and 'large' holes may be identified with 'small' and 'large' spikes. Finally, we discuss several explicit spiky string solutions in AdS3 which, at the leading semiclassical order, display the previously studied finite-gap spectrum. We compute the spectral curves of these strings in the large S limit, finding that they correspond to specific regions of the moduli space of the finite-gap curves. We also explain how 'large' spikes may be used in order to extract a discrete system of degrees of freedom from string theory, which can then be matched with the degrees of freedom of the dual gauge theory operators, and how 'small' spikes are in fact very similar to the Giant Magnons living in R x S2.

The most general AdS$_5 imes Y$ solutions of type IIB string theory that are AdS/CFT dual to superconformal field theories in four dimensions can be fruitfully described in the language of generalized geometry, a powerful hybrid of complex and symplectic geometry. We show that the cone over the compact five-manifold $Y$ is generalized Calabi-Yau and carries a generalized holomorphic Killing vector field $xi$, dual to the R-symmetry. Remarkably, this cone always admits a symplectic structure, which descends to a contact structure on $Y$, with $xi$ as Reeb vector field. Moreover, the contact volumes of $Y$, which can be computed by localization, encode essential properties of the dual CFT, such as the central charge and the conformal dimensions of BPS operators corresponding to wrapped D3-branes. We then define a notion of ``generalized Sasakian geometry'', which can be characterized by a simple differential system of three symplectic forms on a four-dimensional transverse space. The correct Reeb vector field for an AdS$_5$ solution within a given family of generalized Sasakian manifolds can be determined---without the need of the explicit metric---by a variational procedure. The relevant functional to minimize is the type IIB supergravity action restricted to the space of generalized Sasakian manifolds, which turns out to be just the contact volume. We conjecture that this contact volume is equal to the inverse of the trial central charge whose maximization determines the R-symmetry of the dual superconformal field theory. The power of this volume minimization is illustrated by the calculation of the contact volumes for a new infinite family of solutions, in perfect agreement with the results of $a$-maximization in the dual mass-deformed generalized conifold theories.