On world-volume supersymmetry of supermembrane action in static gauge

TL;DR

This work investigates residual 3d world-volume supersymmetry in the static-gauge BST supermembrane and its relation to a possible ${\mathcal N}=1$ 3d supersymmetric derivative expansion. By contrasting the ${\mathcal N}=1$ world-volume action with the ${\cal N}=8$ BST action, the authors identify exact matches only in the special dimensions $D=4$ and $D=5$, while for $D=11$ the two frameworks are inequivalent due to the need for the world-volume fermions to transform as space-time spinors under extended SUSY. They show that a key obstructing feature is the antisymmetric tensor term $U^{ijkl}$ (and its associated ${\epsilon^{abc}}$ structure), which is absent in $D=4,5$ but nontrivial otherwise, and they demonstrate this mismatch explicitly in one-loop world-volume S-matrices. The results clarify how extended world-volume supersymmetry can (or cannot) be realized in static gauge and highlight meaningful dimensional coincidences, informing the construction of consistent effective theories for M2-brane dynamics. Overall, the paper maps the precise conditions under which a 3d ${\cal N}=1$ derivative-expanded membrane action can reproduce or fail to reproduce the BST action and its quantum corrections depending on the target dimension.

Abstract

We review and elaborate on the issue of 3d world volume supersymmetry that appears as a residual part of global target space supersymmetry in the BST supermembrane action. While there is no direct ``spinning membrane'' analog of the world-volume supersymmetric spinning string action that could be obtained by coupling $D$ copies of 3d scalar multiplet to 3d supergravity, we discuss how one may construct an $N=1$ 3d supersymmetric analog of the derivative expansion of the bosonic membrane action in static gauge. We compare the resulting $N=1$ supersymmetric action for eight 3d scalar multiplets to the $N=8$ 3d supersymmetric action describing the $D=11$ supermembrane in the static gauge. The two actions are not equivalent which is related to the fact that the full $N=8$ supersymmetry of the static-gauge $D=11$ supermembrane action can be realised only if the fermions are described by an $SO(8)$ spinor rather than vector. The two actions are still directly related in special dimensions $D=4$ and 5. We also compute the one-loop world-volume scattering amplitudes for the two theories, finding that they indeed agree for $D=4,5$ but disagree for $D=11$.