Zero modes for the giant magnon

TL;DR

This work explicitly constructs the eight fermionic zero modes of the Hofman–Maldacena giant magnon from the Green–Schwarz action and fixes them with κ-symmetry, yielding a simple zero-mode action after integrating over the worldsheet coordinate. Upon quantization, these modes generate a 16-state short representation of ${\rm SU}(2|2)\times{\rm SU}(2|2)$ with central charges ${\mathbb C}$, ${\mathbb P}$, and ${\mathbb K}$, and the fermionic generators ${\mathcal Q}_{\alpha a}$ and ${\cal S}_{a\alpha}$ are constructed from the zero modes; in the giant magnon regime, ${\cal S}_{a\alpha} \approx -\zeta^{-1}{\mathcal Q}_{\alpha a}$. The analysis also identifies four bosonic zero modes from broken ${\rm SO}(4)$ and translation symmetries, completing the multiplet and illustrating how semiclassical string data realize the ${\rm SU}(2|2)\times{\rm SU}(2|02)$ algebra. These results bolster the link between giant magnon dynamics, integrability structures, and the representation theory underlying the AdS$_5\times$S$^5$ superstring, and point toward possible exact quantization in this background.

Abstract

We explicitly construct the eight fermion zero mode solutions for the Hofman-Maldacena giant magnon. The solutions are naturally gauge fixed under the κ-symmetry. Substituting the solutions back into the Lagrangian leads to a simple expression that can be quantized directly. We also show how to construct the SU(2|2)\times SU(2|2) superalgebra from these zero modes. For completeness we also find the four bosonic zero mode solutions.