Metric 3-Lie algebras for unitary Bagger-Lambert theories

TL;DR

This paper characterizes a class of metric 3-Lie algebras with a maximally isotropic centre, enabling the decoupling of negative-norm states in Bagger-Lambert theories and establishing their connection to familiar supersymmetric gauge theories.

Contribution

It proves a structure theorem for indefinite-signature 3-Lie algebras with a maximally isotropic centre and constructs new examples beyond index 2, linking these algebras to known gauge theories.

Findings

Decoupling of negative-norm states in Bagger-Lambert theories

Construction of new 3-Lie algebra examples beyond index 2

Explicit relationship between 3-Lie algebra data and gauge theory parameters

Abstract

We prove a structure theorem for finite-dimensional indefinite-signature metric 3-Lie algebras admitting a maximally isotropic centre. This algebraic condition indicates that all the negative-norm states in the associated Bagger-Lambert theory can be consistently decoupled from the physical Hilbert space. As an immediate application of the theorem, new examples beyond index 2 are constructed. The lagrangian for the Bagger-Lambert theory based on a general physically admissible 3-Lie algebra of this kind is obtained. Following an expansion around a suitable vacuum, the precise relationship between such theories and certain more conventional maximally supersymmetric gauge theories is found. These typically involve particular combinations of N=8 super Yang-Mills and massive vector supermultiplets. A dictionary between the 3-Lie algebraic data and the physical parameters in the resulting gauge theories will thereby be provided.