[Paper Review] Three lectures on 3-algebras

José Figueroa-O’Farrill|ArXiv.org|Dec 15, 2008

Advanced Topics in Algebra28 references14 citations

TL;DR

This paper introduces metric 3-Lie algebras as a generalization of metric Lie algebras, focusing on their role in three-dimensional superconformal Chern–Simons theories, particularly $N=6$ theories. It establishes a correspondence between such 3-algebras and metric real Lie superalgebras via a construction using a Lie algebra and a unitary representation, providing a Lie-algebraic framework for deconstructing and reconstructing 3-algebras relevant to M-theory compactifications.

ABSTRACT

These notes are based on lectures given in Valencia in October 2008 and in Stockholm in November 2008, in the framework of the Nordita workshop "Geometrical aspects of String Theory". We introduce the notion of a metric 3-Lie algebra and review some of the classification results. We explain the deconstruction of metric 3-Lie algebras in Lie algebraic terms and introduce a general framework in which to describe other 3-algebras of relevance in the description of three-dimensional superconformal Chern-Simons theories, particularly those with N=6. The emphasis throughout is on the general ideas and concrete examples.

Motivation & Objective

• To generalize metric Lie algebras to metric 3-Lie algebras and clarify their role in 3D superconformal field theories.
• To classify metric 3-Lie algebras using Lie algebraic deconstruction techniques.
• To establish a correspondence between $N=6$ Chern–Simons theories and metric real Lie superalgebras with specific odd-odd brackets.
• To provide a systematic framework for constructing and analyzing 3-algebras relevant to M2-brane effective theories.

Proposed method

• Introduce metric 3-Lie algebras via a trilinear bracket, symmetric inner product, and the fundamental identity, generalizing Lie algebra structures.
• Use the Nambu bracket on 3-manifolds as a geometric model for metric 3-Lie algebras.
• Deconstruct metric 3-Lie algebras into a Lie algebra $\mathfrak{g}$ and a unitary representation $V$ via a map $D: V \times V \to \mathfrak{g}$.
• Reconstruct 3-algebras from a Lie algebra $\mathfrak{g}$ and a representation $V$ using the Faulkner construction and sesquilinear maps.
• Show that the fundamental identity in the 3-algebra corresponds to the Jacobi identity in a complex Lie superalgebra $\mathfrak{g}_{\mathbb{C}} \oplus (V \oplus \overline{V})$.
• Establish a one-to-one correspondence between $N=6$ 3-algebras and metric real Lie superalgebras $\mathfrak{g} \oplus [\![V]\!]$ with mixed-type odd-odd brackets.

Experimental results

Research questions

• RQ1 How can metric 3-Lie algebras be systematically classified and related to Lie algebras?
• RQ2 What is the geometric and algebraic structure of the Nambu bracket on 3-manifolds, and how does it relate to 3-Lie algebras?
• RQ3 How can $N=6$ Chern–Simons theories be described using 3-algebras constructed from Lie algebras and unitary representations?
• RQ4 What conditions ensure that the derived 3-algebra satisfies the fundamental identity and metricity?
• RQ5 What is the precise correspondence between $N=6$ 3-algebras and metric Lie superalgebras?

Key findings

• Metric 3-Lie algebras generalize metric Lie algebras by replacing the bilinear bracket with a trilinear, totally antisymmetric bracket satisfying metricity and the fundamental identity.
• The Nambu bracket on a 3-manifold with a volume form provides a natural geometric example of a metric 3-Lie algebra.
• Every metric 3-Lie algebra can be deconstructed into a Lie algebra $\mathfrak{g}$ and a unitary representation $V$ via a map $D: V \times V \to \mathfrak{g}$, with $D$ satisfying specific symmetry and compatibility conditions.
• The fundamental identity in the 3-algebra is equivalent to the Jacobi identity in a complex Lie superalgebra $\mathfrak{g}_{\mathbb{C}} \oplus (V \oplus \overline{V})$, with nonvanishing odd-odd brackets only between $V$ and $\overline{V}$.
• There is a one-to-one correspondence between $N=6$ 3-algebras and metric real Lie superalgebras $\mathfrak{g} \oplus [\![V]\!]$, where $V$ is a complex unitary representation of $\mathfrak{g}$ and the only nonvanishing odd-odd brackets are of mixed type.
• The $N=8$ theory corresponds to a special case where the 3-algebra satisfies additional symmetry conditions, and the associated Lie superalgebra is $\mathfrak{psu}(n|n)$, the real form of $A(n-1,n-1)$.