Enhanced N=8 Supersymmetry of ABJM Theory on R(8) and R(8)/Z(2)

TL;DR

The paper proves that ABJM theories with gauge group $U(N)\times U(M)$ exhibit enhanced $N=8$ supersymmetry and $SO(8)$ R-symmetry at Chern-Simons levels $k=1$ and $k=2$ by employing monopole operators within a hermitian 3-algebra framework. The authors recast ABJM in a BLG-like (trial) formulation, show that extra terms vanish when integrating out CS fields at $k=1,2$, and identify an additional $N=2$ supersymmetry that combines with the manifest $N=6$ to yield full on-shell $N=8$ symmetry, with the scalar potential becoming $SO(8)$ invariant. Central to the construction are rank-2 monopole operators enabling gauge-covariant lifting of ABJM fields and a set of algebraic identities derived from the field-strength and equations of motion that ensure closure of the extended supersymmetry. The results illuminate the symmetry structure of ABJM theories, justify the special status of $k=1,2$, connect ABJM to BLG through hermitian 3-algebras and triality, and have implications for AdS$_4$/CFT$_3$ holography and the counting of degrees of freedom in M2-brane dynamics.

Abstract

The ABJM theory refers to superconformal Chern-Simons-matter theory with product gauge group U(L)xU(R) and level +k, -k, respectively. The theory is a candidate for worldvolume dynamics of M2-branes sitting at C(4)/Z(k)k. By utilizing monopole operators, we prove that ABJM theory gets enhanced N=8 supersymmetry and SO(8) R-symmetry at Chern-Simons levels k=1,2. We first show that the ABJM Lagrangian can be written in a manifestly SO(8) invariant form up to certain extra terms. We then show that upon integrating out Chern-Simons gauge fields these extra terms vanish precisely at levels k=1,2. Utilizing monopole operators at these levels, we identify new N=2 supersymmetry. We demonstrate that they combine with the manifest N=6 supersymmetry to close on-shell on N=8 supersymmetry. We finally show that the ABJM scalar potential is SO(8) invariant.