On the Lie-Algebraic Origin of Metric 3-Algebras

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–Sämann, 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–Sämann 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.