Single spike solutions for strings onS2andS3

Abstract

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.

Spiky strings, giant magnons, andβdeformations

“…Recently a new analysis of the class of spiky string appeared, in [25] infinitely wound string solutions with single spikes on S 2 and S 3 were found. It can be shown that these solutions can be found in a certain limit of the parameters of a general rotating rigid string.…”

On elliptic string solutions in AdS3 and dS3

“…It would be interesting to study the extension to other target space geometries, such as the sphere. Spiky string solutions are known to exist on the sphere [39,45], thus it is very probable that there is an analogous treatment for them. In higher dimensional symmetric spaces, Pohlmeyer reduction results in multi-component generalizations of the sinh-or cosh-Gordon equations.…”

Neumann and Neumann-Rosochatius integrable systems from membranes onAdS4×S7

“…In particular, our Lagrangian (3.33), being completely analogous to the one given in (2.26) of [9], should lead to the same energy-charge relation for the giant magnon solution with two angular momenta (see also [27,34]). Moreover, the single spike solutions of [41] can be reproduced also from membranes on AdS 4 × S 7 [42]. In addition, one can consider the correspondence between the Neumann and Neumann-Rosochatius integrable systems arising from membranes and the continuous limit of integrable spin chains at the level of actions, as is done in [43].…”