Constraining Maximally Supersymmetric Membrane Actions

TL;DR

This paper proves that under certain positivity conditions, the only maximally supersymmetric membrane actions in three dimensions are based on known algebraic structures, confirming their uniqueness.

Contribution

It establishes a classification result showing that only known algebraic structures can produce such supersymmetric membrane actions under the given assumptions.

Findings

Only the four-dimensional algebra yields non-trivial solutions.

Direct sums of the four-dimensional algebra are also valid.

No other algebraic structures satisfy the positivity and compatibility conditions.

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.