Zero Modes for the D=11 Membrane and Five-Brane

TL;DR

This work analyzes the zero-mode structure of the extremal $D=11$ membrane and five-brane within eleven-dimensional supergravity to shed light on their quantum behavior and duality with $D=10$ Type IIA theory. By deriving the effective worldvolume actions from the $D=11$ Lagrangian and classifying normalizable zero modes, the authors show that the membrane possesses 8 bosonic zero modes corresponding to translations, and the five-brane also has 8 bosonic zero modes, comprising 5 translational modes and 3 from a self-dual (or anti-self-dual) worldvolume 3-form. The resulting effective actions reproduce the expected quadratic form for the collective coordinates, consistent with general $p$-brane theory and supporting the $D=11$–$D=10$ duality. These findings lay groundwork for a more complete quantum treatment of these branes and highlight their role in the M-theory/IIA correspondence.

Abstract

There exist extremal p-brane solutions of $D\!=\!11$ supergravity for p=2~and~5. In this paper we investigate the zero modes of the membrane and the five-brane solutions as a first step toward understanding the full quantum theory of these objects. It is found that both solutions possess the correct number of normalizable zero modes dictated by supersymmetry.

Figures (1)

• Figure 1: Causal Structure of the five-brane. Each point represents the product of a four-sphere with a five-plane. The dashed line is a spacelike hypersurface.