Finite-size effects of Membranes on AdS_4 x S_7

TL;DR

The paper advances the AdS$_4$/CFT$_3$ correspondence by analyzing semi-classical M2-brane dynamics on $AdS_4\times S^7$, identifying GM- and SS-like states via a Neumann-Rosochatius reduction and establishing a direct mapping to the complex sine-Gordon model. This NR$\to$CSG correspondence allows explicit construction of finite-size membrane solutions and their energy-charge corrections, extending known finite-size effects from strings to membranes. The key result is a concrete, parameter-rich bridge between NR integrable dynamics and CSG, enabling precise dispersion corrections and a framework for exploring more general membrane configurations. The findings illuminate how finite-size effects encode dual CFT data in the M-theory regime and offer pathways for studying scattering and multi-spin generalizations in $AdS_4\times S^7$.

Abstract

We consider semi-classical solution of membranes on the AdS_4 x S^7. This is supposed to be dual to the N=6 super Chern-Simons theory with k=1 in a planar limit recently proposed by Aharony, Bergmann, Jafferis, and Maldacena (ABJM). We have identified giant magnon and single spike states on the membrane by reducing them to the Neumamm - Rosochatius integrable system. We also connect these to the complex sine-Gordon integrable model. Based on this approach, we find finite-size membrane solutions and obtain their images in the complex sine-Gordon system along with the leading finite-size corrections to the energy-charge relations.