On world-volume supersymmetry of supermembrane action in static gauge

TL;DR

This paper explores the world-volume supersymmetry of the supermembrane in static gauge, comparing different supersymmetric actions and analyzing their equivalence and scattering amplitudes in various dimensions.

Contribution

It constructs an N=1 supersymmetric analog of the membrane action in static gauge and compares it to the full N=8 supersymmetric action, analyzing their relations and differences.

Findings

The N=1 and N=8 supersymmetric actions are not equivalent in general.

The actions agree in dimensions D=4 and D=5.

One-loop scattering amplitudes match in D=4 and D=5 but differ in D=11.

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 $D=11$.