Giant Magnons in AdS4 x CP3: Embeddings, Charges and a Hamiltonian

TL;DR

The paper investigates giant magnons in AdS$_{4}\times CP^{3}$ by examining their embeddings into CP$^{3}$ subspaces (CP$^{1}$, RP$^{2}$, RP$^{3}$) and by surveying larger subspaces, clarifying geometric constraints that govern their existence. It identifies the fluctuation Hamiltonian $\Delta-(J_{1}-J_{4})/2$ for small excitations about the vacuum and uses it to derive dispersion relations for several inequivalent magnons, including single-charge and dyonic cases, with distinct mass spectra arising from AdS$_{4}$ and CP$^{3}$ fluctuations. The work also discusses finite-$J$ corrections, the embedding of finite-$J solutions, and connections to the algebraic curve, while highlighting subtleties in embedding larger subspaces such as CP$^{2}$ and $S^{2}\times S^{2}$. Overall, the results illuminate how CP$^{3}$ geometry shapes the magnon spectrum and provide a framework for comparing string solutions to spin-chain excitations in the ABJM setting. The study thus advances understanding of semiclassical string dynamics in AdS$_{4}\times CP^{3}$ and informs future quantum analyses via Pohlmeyer reduction and algebraic-curve methods.

Abstract

This paper studies giant magnons in CP3, which in all known cases are old solutions from S5 placed into two- and three-dimensional subspaces of CP3, namely CP1, RP2 and RP3. We clarify some points about these subspaces, and other potentially interesting three- and four-dimensional subspaces. After confirming that E-(J1-J4)/2 is a Hamiltonian for small fluctuations of the relevant 'vacuum' point particle solution, we use it to calculate the dispersion relation of each of the inequivalent giant magnons. We comment on the embedding of finite-J solutions, and use these to compare string solutions to giant magnons in the algebraic curve.

Figures (1)

• Figure 1: Two giant magnons are shown (in red) on the unit sphere $S^{2}$ (left), on $RP^{2}$ (centre, drawn here as half a sphere) and on $CP^{1}$, a sphere of radius $\frac{1}{2}$ (right). In all cases they have $p_{1}=\tfrac{1}{2}$ and $p_{2}=\pi-\tfrac{1}{2}$, which leads to a closed string in the $RP^{2}$ case, but not in the $S^{2}$ or $CP^{1}$ cases. In both the $RP^{2}$ and $CP^{1}$ cases, the equator is of length $\pi$, and we parameterise it by $\beta\in[0,\pi]$. The magnon with $p_{1}=\frac{1}{2}$ spans $\Delta\beta=\frac{1}{2}$ in the $RP^{2}$ case, but only $\Delta\beta=\frac{1}{4}$ in the $CP^{1}$ case. On $CP^{1}$ we have also drawn a third magnon (in blue) with $p_{3}=1$, which spans the same length of equator $\Delta\beta=\frac{1}{2}$ as does the $p_{1}$ magnon on $RP^{2}$.