Classification of N=6 superconformal theories of ABJM type

TL;DR

The paper classifies ${\mathcal N}=6$ superconformal theories of ABJM type by deriving a generator-level condition on the gauge group and matter content that yields supersymmetry enhancement. Through a Dynkin-diagram analysis, it shows that only a narrow set of gauge groups with bifundamental matter survive: $SU(n)\times U(1)$, $Sp(n)\times U(1)$, $SU(n)\times SU(n)$, and $SU(n)\times SU(m)\times U(1)$ (with possible extra $U(1)$ factors), and that reducible representations do not enlarge the landscape. It provides explicit constructions for the allowed irreducible cases—$SU(m)\times SU(n)\times U(1)$ and $Sp(n)\times U(1)$—and clarifies how abelian factors can be incorporated via a level matrix, ensuring the required antisymmetry of the $f$-tensor. The work also yields an independent route to the BLG three-algebra classification and discusses the (rare) route to ${\mathcal N}=8$ enhancement, identifying the unique BLG-type realization within this framework.

Abstract

Studying the supersymmetry enhancement mechanism of Aharony, Bergman, Jafferis and Maldacena, we find a simple condition on the gauge group generators for the matter fields. We analyze all possible compact Lie groups and their representations. The only allowed gauge groups leading to the manifest N=6 supersymmetry are, up to discrete quotients, SU(n) x U(1), Sp(n) x U(1), SU(n) x SU(n), and SU(n) x SU(m) x U(1) with possibly additional U(1)'s. Matter representations are restricted to be the (bi)fundamentals. As a byproduct we obtain another proof of the complete classification of the three algebras considered by Bagger and Lambert.

Figures (8)

• Figure 1: Choice of the states for \ref{['bra-eq']}
• Figure 2: Three adjacent roots
• Figure 3: Choice of states for \ref{['eq_terminal']}
• Figure 4: Branched Dynkin diagram
• Figure 5: Choice of states for Sec. \ref{['short']}