Constraining Maximally Supersymmetric Membrane Actions

TL;DR

The paper classifies maximally supersymmetric three-dimensional field theories built from triple-product algebras with a positive-definite metric, showing that only a basic four-dimensional algebra (up to decoupled copies) survives, characterized by a totally antisymmetric 4-form proportional to the Levi-Civita symbol $ε^{μ_1 μ_2 μ_3 μ_4}$. It achieves this by decomposing the structure constants as $f = dx^{D+1} ∧ ψ + φ$ and analyzing the resulting Lie algebra $L$ via the Jacobi identity, distinguishing semi-simple from non-semi-simple cases. In the semi-simple branch, φ vanishes and ψ yields $su(2)$, reproducing the four-dimensional solution; in the non-semi-simple branch, f decomposes into an orthogonal sum of simple 4-forms, hence no new interacting theories arise under the assumptions. The work situates this result as a constraint on Bagger–Gustavsson-type constructions and argues that relaxing positivity or Lagrangian requirements may be necessary to access other maximally supersymmetric membrane theories.

Abstract

We study the recent construction of maximally supersymmetric field theory Lagrangians in three spacetime dimensions that are based on algebras with a triple product. Assuming that the algebra has a positive definite metric compatible with the triple product, we prove that the only non-trivial examples are either the well known case based on a four dimensional algebra or direct sums thereof.