We study solutions for rigidly rotating strings on a two sphere. Among them we find two limiting cases that have a particular interest, one is the already known giant magnon and the other we call the single spike solution. The limiting behavior of this last solution is a string infinitely wrapped around the equator. It differs from that solution by the existence of a single spike of heightθ that points toward the north pole. We study its properties and compute its energy E and angular momentum J as a function ofθ. We further generalize the solution by adding one angular momentum to obtain a solution on S 3 . We find a spin chain interpretation of these results in terms of free fermions and the Hubbard model but the exact relation with the same models derived from the field theory is not clear.

The AdS/CFT correspondence relates Wilson loops in N = 4 SYM theory to minimal area surfaces in AdS 5 space. In this paper we consider the case of Euclidean flat Wilson loops which are related to minimal area surfaces in Euclidean AdS 3 space. Using known mathematical results for such minimal area surfaces we describe an infinite parameter family of analytic solutions for closed Wilson loops. The solutions are given in terms of Riemann theta functions and the validity of the equations of motion is proven based on the trisecant identity. The world-sheet has the topology of a disk and the renormalized area is written as a finite, one-dimensional contour integral over the world-sheet boundary. An example is discussed in detail with plots of the corresponding surfaces. Further, for each Wilson loops we explicitly construct a one parameter family of deformations that preserve the area. The parameter is the so called spectral parameter. *

We study a class of classical solutions for closed strings moving in AdS 3 × S 1 ⊂ AdS 5 × S 5 with energy E and spin S in AdS 3 and angular momentum J and winding m in S 1 . They have rigid shape with n spikes in AdS 3 . We find that when J or m are non-zero, the spikes do not end in cusps. We consider in detail a special large n limit in which S ∼ n 2 , J ∼ n, i.e. S ≫ J ≫ 1, withn staying finite. In that limit the spiky spinning string approaches the boundary of AdS 5 . We show that the corresponding solution can be interpreted as describing a periodic-spike string moving in AdS 3 -pp-wave ×S 1 background. The resulting expression for the string energy should represent a strong-coupling prediction for anomalous dimension of a class of dual gauge theory states in a particular thermodynamic limit of the SL(2, R) spin chain.

We apply the dressing method to a string solution given by a static string wrapped around the equator of a three-sphere and find that the result is the single spike solution recently discussed in the literature. Further application of the method allows the construction of solutions with multiple spikes. In particular we construct the solution describing the scattering of two single spikes and compute the scattering phase shift. As a function of the dressing parameters, the result is exactly the same as the one for the giant magnon, up to non-logarithmic terms. This suggests that the single spikes should be described by an integrable spin chain closely related to the one associated to the giant magnons. The field theory interpretation of such spin chain however is still unclear.

We describe new solutions for open string moving in AdS 5 and ending in the boundary, namely dual to Wilson loops in N = 4 SYM theory. First we introduce an ansatz for Euclidean curves whose shape contains an arbitrary function. They are BPS and the dual surfaces can be found exactly.After an inversion they become closed Wilson loops whose expectation value is W = exp(− √ λ).After that we consider several Wilson loops for N = 4 SYM in a pp-wave metric and find the dual surfaces in an AdS 5 pp-wave background. Using the fact that the pp-wave is conformally flat, we apply a conformal transformation to obtain novel surfaces describing strings moving in AdS space in Poincare coordinates and dual to Wilson loops for N = 4 SYM in flat space.