ABJM

ABJM theory, named after Aharony, Bergman, Jafferis, and Maldacena, is a three-dimensional N=6 superconformal Chern-Simons-matter theory proposed in 2008. It describes the low-energy dynamics of N coincident M2-branes on a C^4/Z_k singularity and is characterized by the gauge group U(N)_k × U(N)_{-k}, with Chern-Simons levels k and −k. The matter content consists of four complex scalar fields and their fermionic superpartners, arranged in bifundamental representations under the two gauge groups. The theory possesses conformal invariance and an SU(4) R-symmetry, together forming the superconformal algebra OSp(6|4). For k=1 or 2, supersymmetry enhances to N=8 and the R-symmetry to SO(8).

Classically, the Lagrangian contains Chern-Simons terms, kinetic terms for matter, and a sextic scalar potential that preserves the N=6 supersymmetry. The moduli space for N M2-branes is the symmetric product (C^4/Z_k)^N / S_N, corresponding to the possible brane positions on the orbifold.

ABJM provides a concrete realization of AdS4/CFT3: in the large-N limit with k fixed, the dual description is M-theory on AdS4 × S7/Z_k; in the limit N >> k^2, it reduces to type IIA string theory on AdS4 × CP3 with 't Hooft coupling λ = N/k.

The theory is integrable in the planar limit and has been extensively studied using localization and other techniques to probe three-dimensional conformal dynamics, dualities, and M-theory.