Superalgebra realization of the 3-algebras in ${\cal N}=6, 8$N=6,8 Chern-Simons-matter theories

Abstract

We use superalgebras to realize the 3-algebras used to construct \documentclass[12pt]{minimal}\begin{document}${\cal N}=6, 8$\end{document}N=6,8 Chern-Simons-matter (CSM) theories. We demonstrate that the superalgebra realization of the 3-algebras provides a unified framework for classifying the gauge groups of the \documentclass[12pt]{minimal}\begin{document}${\cal N}\ge 5$\end{document}N≥5 theories based on 3-algebras. Using this realization, we rederive the ordinary Lie algebra construction of the general \documentclass[12pt]{minimal}\begin{document}${\cal N}=6$\end{document}N=6 CSM theory from its 3-algebra counterpart and reproduce all known examples as well. In particular, we explicitly construct the Nambu 3-bracket in terms of a double graded commutator of PSU(2|2). The \documentclass[12pt]{minimal}\begin{document}${\cal N}=8$\end{document}N=8 theory of Bagger, Lambert and Gustavsson (BLG) with SO(4) gauge group is constructed by using several different ways. A quantization scheme for the 3-brackets is proposed by promoting the double graded commutators as quantum mechanical double graded commutators.