On the Lie-Algebraic Origin of Metric 3-Algebras

De Medeiros, Paul F., Figueroa-O'Farrill, José, Méndez-Escobar, Elena and Ritter, Patricia 2009. On the Lie-Algebraic Origin of Metric 3-Algebras. Communications in Mathematical Physics 290 (3) , pp. 871-902. 10.1007/s00220-009-0760-1

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Abstract

Since the pioneering work of Bagger–Lambert and Gustavsson, there has been a proliferation of three-dimensional superconformal Chern–Simons theorieswhose 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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Schools > Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	Springer
ISSN:	0010-3616
Last Modified:	19 Mar 2016 23:07
URI:	https://orca.cardiff.ac.uk/id/eprint/39668

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