Lie 3-Algebra and Multiple M2-branes

TL;DR

This paper explores the structure of Lie 3-algebras, their representations, and applications to multiple M2-branes, revealing new algebraic structures and their relevance to membrane physics.

Contribution

It provides a detailed analysis of Lie 3-algebras, introduces new algebraic structures, and discusses their applications in M2-brane models and membrane physics.

Findings

Discovery of new Lie 3-algebras not previously known

Development of tensor product methods for higher-dimensional representations

Application of Lie 3-algebras to membrane physics models

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.