M2-branes, 3-Lie algebras and Plücker relations

Abstract

We find that the structure constants 4-form of a metric 3-Lie algebra is the sum of the volume forms of orthogonal 4-planes proving a conjecture in math/0211170. In particular, there is no metric 3-Lie algebra associated to a (N) Lie algebra for 2$">N>2. We examine the implication of this result on the existence of a multiple M2-brane theory based on metric 3-Lie algebras.