Lie 3-Algebra and Multiple M2-branes

TL;DR

This work investigates Lie $3$-algebras in the context of multiple M2-branes, focusing on the fundamental identity and its links to Nambu–Poisson structures. It introduces new finite-dimensional algebras with non-positive-definite metrics, develops cubic-matrix representations and higher-dimensional constructions, and analyzes irreducibility and projector-based extensions. The Basu–Harvey funnel and the Bagger–Lambert action are examined through the lens of $3$-algebra structure, showing that the FI underpins gauge symmetry and BPS conditions independent of a specific representation. The study highlights the scarcity of positive-definite finite-dimensional examples and discusses the physical implications for M2–M5 systems, suggesting directions for identifying broader algebraic frameworks in M-theory. These insights advance the algebraic toolkit for modeling non-Abelian M2-brane dynamics and their connections to higher-dimensional brane configurations.

Abstract

Motivated by the recent proposal of an N=8 supersymmetric action for multiple M2-branes, we study the Lie 3-algebra in detail. In particular, we focus on the fundamental identity and the relation with Nambu-Poisson bracket. Some new algebras not known in the literature are found. Next we consider cubic matrix representations of Lie 3-algebras. We show how to obtain higher dimensional representations by tensor products for a generic 3-algebra. A criterion of reducibility is presented. We also discuss the application of Lie 3-algebra to the membrane physics, including the Basu-Harvey equation and the Bagger-Lambert model.