Finite size Giant Magnons in the string dual of N=6 superconformal Chern-Simons theory

TL;DR

This paper constructs an exact finite-size Giant Magnon in the $AdS_4\times CP^3$ background, within the $SU(2)\times SU(2)$ sector relevant to ABJM theory, showing the magnon corresponds to two oppositely oriented magnons on the two $S^2$ spheres. The authors solve the classical string equations on a restricted $R\times S^2\times S^2$ subspace, obtaining a solution expressed through Jacobi elliptic functions with turning points $z_{\max}$ and $z_{\min}$ and deriving explicit relations for the energy $\Delta$, angular momentum $J$, and momentum $p$ in terms of elliptic integrals $K(\nu)$, $E(\nu)$ and $\Pi(\cdot;\nu)$. In the magnon limit of large $\Delta$ and $J$ with $\Delta-J$ finite, they recover the infinite-volume dispersion $\Delta-J \to 2\sqrt{2\lambda}\,|\sin(p/2)|$ and identify the leading finite-size correction, which is exponentially suppressed as $\Delta-J \simeq 2\sqrt{2\lambda}\left(|\sin(p/2)| - 4|\sin(p/2)|^{3} e^{-\Delta/(\sqrt{2\lambda}\,|\sin(p/2)|)}\right)$. The turning points scale with $p$ (e.g., $z_{\max}\to|\sin(p/2)|$) and the solution can be adapted to orbifold settings, where $p$ is quantized as $p=2\pi m/M$. This work extends finite-size analyses known from AdS$_5$/CFT$_4$ to the ABJM context and provides a concrete, exact classical string description including orbifold considerations.

Abstract

We find the exact solution for a finite size Giant Magnon in the $SU(2)\times SU(2)$ sector of the string dual of the $\mathcal{N}=6$ superconformal Chern-Simons theory recently constructed by Aharony, Bergman, Jafferis and Maldacena. The finite size Giant Magnon solution consists of two magnons, one in each $SU(2)$. In the infinite size limit this solution corresponds to the Giant Magnon solution of arXiv:0806.4959. The magnon dispersion relation exhibits finite-size exponential corrections with respect to the infinite size limit solution.