M2-branes, 3-Lie Algebras and Plucker relations

TL;DR

The paper proves that the structure constants of a metric 3-Lie algebra with Euclidean signature decompose as a sum of volume forms of mutually orthogonal 4-planes, i.e., $F = \sum_r \mu_r vol(V_r)$. The approach uses associated metric Lie algebras $a_{[2]}(X)$ to translate the Jacobi constraints into Plücker-type relations, distinguishing cases with and without bi-invariant 4-forms and showing that, in all scenarios, the nontrivial solutions reduce to sums of orthogonal simple 4-forms. A full classification shows all metric Lie algebras are isomorphic to $⊕^ℓ u(1) ⊕ ss$, which underpins the main result; the implications for multiple M2-brane theories are significant, as the expected $U(N)$ gauge symmetry for coincident D2-branes cannot arise from Euclidean metric 3-Lie algebras for $N>2$. The work also discusses extensions to higher $k$-Lie algebras and the Lorentzian case, highlighting avenues where the standard M2-brane constructions may require weakened assumptions or alternative signatures for consistency.

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.