Notes on Supersymmetry Enhancement of ABJM Theory

TL;DR

This work addresses the long-standing question of supersymmetry enhancement in ABJM theory by identifying an extra ${\ncal N}=2$ SUSY generated by a monopole (t\$Hooft) operator $T$ that can pair with the innate ${\ncal N}=6$ SUSY to yield ${\ncal N}=8$ for certain gauge groups. The authors develop a minimal ${\ncal N}=2$ CS-matter model and demonstrate the enhancement for $U(1)\times U(1)$, $SU(2)\times SU(2)$, and $U(2)\times U(2)$, with the enhancement in the abelian case restricted to $k=1,2$ due to ${\bf Z}_k$ orbifolding, while the nonabelian cases exhibit enhancement more broadly. They then extend the analysis to ABJM theory with $U(2)\times U(2)$, providing explicit ${\ncal N}=2$ transformation rules that incorporate the monopole operator and showing ${\ncal N}=8$ for $k=1,2$, and they propose a general formulation for ${U(N)\times U(N)}$ involving a differential equation for the monopole operator. The results illuminate the mechanism of SUSY enhancement in M2-brane worldvolume theories and offer a path to generalize to arbitrary gauge groups, with potential implications for dual gravity descriptions and nonperturbative operator dynamics.

Abstract

We study the supersymmetry enhancement of ABJM theory. Starting from a ${\cal N}=2$ supersymmetric Chern-Simons matter theory with gauge group U(2)$\times$U(2) which is a truncated version of the ABJM theory, we find by using the monopole operator that there is additional ${\cal N}=2$ supersymmetry related to the gauge group. We show this additional supersymmetry can combine with ${\cal N}=6$ supersymmetry of the original ABJM theory to an enhanced ${\cal N}=8$ SUSY with gauge group U(2)$\times$U(2) in the case $k=1,2$. We also discuss the supersymmetry enhancement of the ABJM theory with U($N$)$\times$U($N$) gauge group and find a condition which should be satisfied by the monopole operator.