Spiky strings and the AdS/CFT correspondence
In this dissertation, we explore some aspects of semiclassical type IIB string theory on AdS3 × 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 × 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 × 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 × S2.