On the Lie-algebraic origin of metric 3-algebras

TL;DR

This paper reveals that certain metric 3-algebras used in superconformal Chern-Simons theories originate from pairs of metric Lie algebras and faithful unitary representations, unifying various algebraic structures in the field.

Contribution

It demonstrates the Lie-algebraic origin of metric 3-algebras, including those in superconformal theories, and provides a unified construction from pairs of Lie algebras and representations.

Findings

Real 3-algebras are characterized by Cherkis-Saemann algebras.

Complex and quaternionic 3-algebras generalize Bagger-Lambert structures.

The construction links 3-algebras to metric Lie algebras and unitary representations.

Abstract

Since the pioneering work of Bagger-Lambert and Gustavsson, there has been a proliferation of three-dimensional superconformal Chern-Simons theories whose main ingredient is a metric 3-algebra. On the other hand, many of these theories have been shown to allow for a reformulation in terms of standard gauge theory coupled to matter, where the 3-algebra does not appear explicitly. In this paper we reconcile these two sets of results by pointing out the Lie-algebraic origin of some metric 3-algebras, including those which have already appeared in three-dimensional superconformal Chern-Simons theories. More precisely, we show that the real 3-algebras of Cherkis-Saemann, which include the metric Lie 3-algebras as a special case, and the hermitian 3-algebras of Bagger-Lambert can be constructed from pairs consisting of a metric real Lie algebra and a faithful (real or complex, respectively) unitary representation. This construction generalises and we will see how to construct many kinds of metric 3-algebras from pairs consisting of a real metric Lie algebra and a faithful (real, complex or quaternionic) unitary representation. In the real case, these 3-algebras are precisely the Cherkis-Saemann algebras, which are then completely characterised in terms of this data. In the complex and quaternionic cases, they constitute generalisations of the Bagger-Lambert hermitian 3-algebras and anti-Lie triple systems, respectively, which underlie N=6 and N=5 superconformal Chern-Simons theories, respectively. In the process we rederive the relation between certain types of complex 3-algebras and metric Lie superalgebras.