Metric Lie 3-algebras in Bagger-Lambert theory

Abstract

We recast physical properties of the Bagger-Lambert theory, such as shiftsymmetry and decoupling of ghosts, the absence of scale and parity invariance, in Lie 3algebraic 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.

Lorentzian Lie (3-)algebra and toroidal compactification of M/string theory

“…In section 3, we give a detailed study of the constraint from the fundamental identity. Such study for two Lorentzian pairs was made in [11] but we generalize their result by considering an arbitrary number of Lorentzian pairs. We use a strategy to analyze the constraint for the structure constants directly.…”

“…We note that this is the simplest example considered in [11]. For this simplest choice, we see that BLG model gives rise to a free N = 8 supersymmetric massive gauge theory after the Higgs mechanism is used to eliminate the negative-norm fields.…”

“…Such algebras have been considered extensively by de Medeiros et. al [11] when the number of Lorentzian pairs is two. Here we present more straightforward and explicit analysis in terms of the structure constants.…”

“…The simplest class of solutions can be found when f ijk a = 0 for i, j, k ∈ I a . In this case, for this range I a , arbitrary anti-symmetric matrix J ij (i, j ∈ I a ) solves the constraints (this case is a special case of solutions in [11]). We will study the BLG model for this case in section 4.…”

Some results on metric n-Lie algebras

“…Furthermore, they conjectured that there are no other 3-Lie algebras with a positive definite metric except for A 4 and its direct sum, where A 4 is the SO(4)-invariant algebra with 4 generators. In physics, some dynamical systems involve zero-norm generators corresponding to gauge symmetries and negative-norm generators corresponding to ghosts (see [14][15][16]). These messages motivate us to study n-Lie algebras with any metrics or a nondegenerate invariant symmetric bilinear form.…”