Metric Lie 3-algebras in Bagger-Lambert theory

TL;DR

This paper explores the structure and classification of metric Lie 3-algebras relevant to Bagger-Lambert theory, revealing new constructions and analyzing their derivations to understand physical properties like ghost decoupling.

Contribution

It provides a structure theorem for metric Lie 3-algebras of arbitrary signature and classifies those of signature (2,p), introducing a double extension construction.

Findings

Structured metric Lie 3-algebras can be built from simple and one-dimensional algebras.

Classified metric Lie 3-algebras of signature (2,p).

Analyzed derivations preserving the conformal class of the inner product.

Abstract

We recast physical properties of the Bagger-Lambert theory, such as shift-symmetry and decoupling of ghosts, the absence of scale and parity invariance, in Lie 3-algebraic terms, thus motivating the study of metric Lie 3-algebras and their Lie algebras of derivations. We prove a structure theorem for metric Lie 3-algebras in arbitrary signature showing that they can be constructed out of the simple and one-dimensional Lie 3-algebras iterating two constructions: orthogonal direct sum and a new construction called a double extension, by analogy with the similar construction for Lie algebras. We classify metric Lie 3-algebras of signature (2,p) and study their Lie algebras of derivations, including those which preserve the conformal class of the inner product. We revisit the 3-algebraic criteria spelt out at the start of the paper and select those algebras with signature (2,p) which satisfy them, as well as indicate the construction of more general metric Lie 3-algebras satisfying the ghost-decoupling criterion.