Neumann-Rosochatius system for strings in ABJ Model

Abstract

Neumann-Rosochatius system is a well known one dimensional integrable system. We study the rotating and pulsating string in $Ad{S}_4\times {\mathbb{CP}}^3$ with a $B_{\mathrm{NS}}$ holonomy turned on over ${\mathbb{CP}}^1\subset {\mathbb{CP}}^3$, or the so called Aharony-Bergman-Jafferis (ABJ) background. We observe that the string equations of motion in both cases are integrable and the Lagrangians reduce to a form similar to that of deformed Neuman-Rosochatius system. We find out the scaling relations among various conserved charges and comment on the finite size effect for the dyonic giant magnons on $R_t\times {\mathbb{CP}}^3$ with two angular momenta. For the pulsating string we derive the energy as function of oscillation number and angular momenta along ${\mathbb{CP}}^3$.

20 pages, several typos corrected. Better presented. Added references. To appear in JHEP