On the Lie-algebraic origin of metric 3-algebras

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.

29 pages (v4: really final version to appear in CMP. Example 7 has been improved.)

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• Three-Algebras in N = 5, 6 Superconformal Chern-Simons Theories: Representations and Relations
• Quantized Nambu-Poisson Manifolds in a 3-Lie Algebra Reduced Model
• Monopoles, three-algebras and ABJM theories with $\mathbb{N}=5,6,8$ supersymmetry
• Octonions, G_2 and generalized Lie 3-algebras
• Model of M-theory with Eleven Matrices
• Three lectures on 3-algebras
• Unifying N=5 and N=6
• A New Three-Algebra Representation for the {\cal N}=6 su(N)Xsu(N) Superconformal Chern-Simons Theory
• Aspects of U-duality in BLG models with Lorentzian metric 3-algebras
• Squashing Gravity Dual of N=6 Superconformal Chern-Simons Gauge Theory
• N=5 three-algebras and 5-graded Lie superalgebras
• Remark About Scaling Limit of ABJ Theory
• Metric 3-Leibniz algebras and M2-branes