Cartan-Weyl 3-algebras and the BLG Theory II: Strong-Semisimplicity and Generalized Cartan-Weyl 3-algebras

TL;DR

This paper introduces generalized Cartan-Weyl 3-algebras, extending previous structures, to identify suitable Lie 3-algebras for the BLG theory that support fuzzy 3-sphere solutions and maintain unitarity.

Contribution

It proposes a new class of metric Lie 3-algebras, the generalized Cartan-Weyl 3-algebras, with non-abelian Cartan subalgebras, connecting to semisimple Lie algebras for BLG models.

Findings

Generalized Cartan-Weyl 3-algebras include non-abelian Cartan subalgebras.

These algebras are suitable for fuzzy 3-sphere solutions in BLG models.

The reduction condition links these algebras to semisimple Lie algebras.

Abstract

One of the most important questions in the Bagger-Lambert-Gustavsson (BLG) theory of multiple M2-branes is the choice of the Lie 3-algebra. The Lie 3-algebra should be chosen such that the corresponding BLG model is unitary and admits fuzzy 3-sphere as a solution. In this paper we propose another new condition: the Lie 3-algebras of use must be connected to the semisimple Lie algebras describing the gauge symmetry of D-branes via a certain reduction condition. We show that this reduction condition leads to a natural generalization of the Cartan-Weyl 3-algebras introduced in arXiv:1004.1397. Similar to a Cartan-Weyl 3-algebra, a generalized Cartan-Weyl 3-algebra processes a set of step generators characterized by non-degenerate roots. However, its Cartan subalgebra is non-abelian in general. We give reasons why having a non-abelian Cartan subalgebra may be just right to allow for fuzzy 3-sphere solution in the corresponding BLG models. We propose that generalized Cartan-Weyl 3-algebras is the right class of metric Lie 3-algebras to be used in the BLG theory.