Monopoles, three-algebras and ABJM theories with $\N=5,6,8$ supersymmetry

TL;DR

This paper refines the classification of ABJM theories with various supersymmetries by extending three-algebra formulations, analyzing gauge groups, monopole operators, and supersymmetry enhancement mechanisms.

Contribution

It introduces an extended hermitian three-algebra formulation including U(1) factors, classifies allowed gauge groups, and explores monopole operators' role in supersymmetry enhancement.

Findings

Allowed gauge groups are primarily SU(N)×SU(N), U(N)×U(M), and Sp(N)×U(1).

Only one independent Chern-Simons level exists across these gauge groups.

Monopole operators can potentially enhance supersymmetry from N=5 to N=6.

Abstract

We extend the hermitian three-algebra formulation of ABJM theory to include $U(1)$ factors. With attention payed to extra $U(1)$ factors, we refine the classification of $\N=6$ ABJM theories. We argue that essentially the only allowed gauge groups are $SU(N)\times SU(N)$, $U(N)\times U(M)$ and $Sp(N)\times U(1)$ and that we have only one independent Chern-Simons level in all these cases. Our argument is based on integrality of the $U(1)$ Chern-Simons levels and supersymmetry. A relation between monopole operators and Wilson lines in Chern-Simons theory suggests certain gauge representations of the monopole operators. From this we classify cases where we can not expect enhanced $\N=8$ supersymmetry. We also show that there are two equivalent formulations of $\N=5$ ABJM theories, based on hermitian three-algebra and quaternionic three-algebra respectively. We suggest properties of monopoles in $\N=5$ theories and show how these monopoles may enhance supersymmetry from $\N=5$ to $\N=6$.