M2-branes, 3-Lie Algebras and Plucker relations

TL;DR

This paper proves that the structure constants of metric 3-Lie algebras are sums of volume forms of orthogonal 4-planes, showing no such algebra exists for rak{u}(N) with N>2, impacting M2-brane theories.

Contribution

It confirms a conjecture that metric 3-Lie algebras' structure constants are sums of orthogonal 4-plane volume forms, limiting their association with rak{u}(N) for N>2.

Findings

Structure constants are sums of volume forms of orthogonal 4-planes.

No metric 3-Lie algebra corresponds to rak{u}(N) for N>2.

Implications restrict M2-brane theory constructions.

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 $\mathfrak{u}(N)$ Lie algebra for $N>2$. We examine the implication of this result on the existence of a multiple M2-brane theory based on metric 3-Lie algebras.