Lorentzian Lie 3-algebras and their Bagger-Lambert moduli space

TL;DR

The paper classifies Lorentzian metric Lie 3-algebras with invariant inner products and analyzes the Bagger–Lambert moduli space of vacua. It proves that indecomposable Lorentzian 3-algebras are either 1-dimensional, simple, or in bijection with compact real forms of metric semisimple Lie algebras, realized by a basis {u,v} of null directions and a compact semisimple Lie algebra $\mathfrak{g}$ with specific 3-brackets. The moduli space decomposes into a nondegenerate branch, linked to the classical moduli space of 3D $\mathcal{N}=8$ SYM with gauge algebra $\mathfrak{g}$, and a degenerate branch, governed by maximal $\Omega$-isotropic subspaces corresponding to compact Riemannian symmetric spaces; both branches are analyzed under gauge equivalence via $\mathrm{Ad}\,V$. The authors derive the asymptotic behavior of these branches, demonstrating models with the expected $N^{3/2}$ scaling by suitable choices of $\mathfrak{g}$, thus supporting the viability of Lorentzian Bagger–Lambert constructions for large stacks of M2-branes. They also provide explicit descriptions of automorphism groups and the structure of maximal abelian subalgebras, tying the moduli spaces to Weyl-group quotients and symmetric-space geometry. Overall, the work clarifies the landscape of Lorentzian 3-algebras in M-theory contexts and highlights concrete pathways to realize the desired large-$N$ scaling in the Bagger–Lambert framework.

Abstract

We classify Lie 3-algebras possessing an invariant lorentzian inner product. The indecomposable objects are in one-to-one correspondence with compact real forms of metric semisimple Lie algebras. We analyse the moduli space of classical vacua of the Bagger-Lambert theory corresponding to these Lie 3-algebras. We establish a one-to-one correspondence between one branch of the moduli space and compact riemannian symmetric spaces. We analyse the asymptotic behaviour of the moduli space and identify a large class of models with moduli branches exhibiting the desired N^{3/2} behaviour.

Figures (2)

• Figure 1: Average rank as a function of dimension for compact semisimple Lie algebras
• Figure 2: Average $\dim\mathfrak{p}$ as a function of $\dim\mathfrak{g}$, for $(\mathfrak{g},\mathfrak{p}^\perp)$ a symmetric split

Theorems & Definitions (21)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4: LingSimple
• Lemma 5
• Lemma 6
• proof