ABJM theory



\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory




String theory



The ABJM model (ABJM 08) is an 𝒩=6\mathcal{N} = 6 3d superconformal gauge field theory involving Chern-Simons theory with gauge group SU(N) and coupled to matter fields. For Chern-Simons level kk it is supposed to describe the worldvolume theory of NN coincident black M2-branes at an /k\mathbb{Z}/k-cyclic group orbifold singularity with near-horizon geometry AdS 4×S 7/(/k)AdS_4 \times S^7/(\mathbb{Z}/k) (see at M2-branes – As a black brane).

ddNNsuperconformal super Lie algebraR-symmetryblack brane worldvolume
superconformal field theory
via AdS-CFT
A3A\phantom{A}3\phantom{A}A2k+1A\phantom{A}2k+1\phantom{A}AB(k,2)\phantom{A}B(k,2) \simeq osp(2k+1|4)A(2k+1 \vert 4)\phantom{A}ASO(2k+1)A\phantom{A}SO(2k+1)\phantom{A}
A3A\phantom{A}3\phantom{A}A2kA\phantom{A}2k\phantom{A}AD(k,2)\phantom{A}D(k,2)\simeq osp(2k|4)A(2k \vert 4)\phantom{A}ASO(2k)A\phantom{A}SO(2k)\phantom{A}M2-brane
3d superconformal gauge field theory
A4A\phantom{A}4\phantom{A}Ak+1A\phantom{A}k+1\phantom{A}AA(3,k)𝔰𝔩(4|k+1)A\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}AU(k+1)A\phantom{A}U(k+1)\phantom{A}D3-brane
4d superconformal gauge field theory
A6A\phantom{A}6\phantom{A}AkA\phantom{A}k\phantom{A}AD(4,k)\phantom{A}D(4,k) \simeq osp(8|2k)A(8 \vert 2k)\phantom{A}ASp(k)A\phantom{A}Sp(k)\phantom{A}M5-brane
6d superconformal gauge field theory

(Shnider 88, also Nahm 78, see Minwalla 98, section 4.2)

For k=2k = 2 the supersymmetry of the ABJM model increases to 𝒩=8\mathcal{N} = 8. For k=2k = 2 and N=2N = 2 the ABJM model reduces to the BLG model (ABJM 08, section 2.6).

Due to the matter coupling, the ABJM model is no longer a topological field theory as pure Chern-Simons is, but it is still a conformal field theory. As such it is thought to correspond under AdS-CFT duality to M-theory on AdS4 ×\times S7//k\mathbb{Z}/k (see also MFFGME 09).

Notice that the worldvolume SU(N)SU(N) gauge group enhancement at an k\mathbb{Z}_k-ADE singularity is akin to the gauge symmetry enhancement of the effective field theory for M-theory on G2-manifolds at the same kind of singularities (see at M-theory on G2-manifolds – Nonabelian gauge groups).

More generally, classification of the near horizon geometry of smooth (i.e. non-orbifold) 12\geq \tfrac{1}{2} BPS black M2-brane-solutions of the equations of motion of 11-dimensional supergravity shows that these are the Cartesian product AdS 4×(S 7/G)AdS_4 \times (S^7/G) of 4-dimensional anti de Sitter spacetime with a 7-dimensional spherical space form S 7/G^S^7/{\widehat{G}} with spin structure and N4N \geq 4, for G^\widehat{G} a finite subgroup of SU(2) (MFFGME 09, see here).

NN Killing spinors on
spherical space form S 7/G^S^7/\widehat{G}
AAG^=\phantom{AA}\widehat{G} =spin-lift of subgroup of
isometry group of 7-sphere
3d superconformal gauge field theory
on back M2-branes
with near horizon geometry AdS 4×S 7/G^AdS_4 \times S^7/\widehat{G}
AAN=8AA\phantom{AA}N = 8\phantom{AA}AA 2\phantom{AA}\mathbb{Z}_2cyclic group of order 2BLG model
AAN=7AA\phantom{AA}N = 7\phantom{AA}
AAN=6AA\phantom{AA}N = 6\phantom{AA}AA k>2\phantom{AA}\mathbb{Z}_{k\gt 2}cyclic groupABJM model
AAN=5AA\phantom{AA}N = 5\phantom{AA}AA2D k+2\phantom{AA}2 D_{k+2}
2T2 T, 2O2 O, 2I2 I
binary dihedral group,
binary tetrahedral group,
binary octahedral group,
binary icosahedral group
(HLLLP 08a, BHRSS 08)
AAN=4AA\phantom{AA}N = 4\phantom{AA}A2D k+2\phantom{A}2 D_{k+2}
2O2 O, 2I2 I
binary dihedral group,
binary octahedral group,
binary icosahedral group
(HLLLP 08b, Chen-Wu 10)


AdS/CFT duality

Under holographic duality supposed to be related to M-theory on AdS 4×S 7/ kAdS_4 \times S^7 / \mathbb{Z}_k.

Boundary conditions

Discussion of boundary conditions of the BLG model, leading to brane intersection with M-wave, M5-brane and MO9-brane is in (Chu-Smith 09, BPST 09).

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
(D=2n)(D = 2n)type IIA\,\,
D0-brane\,\,BFSS matrix model
D4-brane\,\,D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane\,\,D=7 super Yang-Mills theory
(D=2n+1)(D = 2n+1)type IIB\,\,
D1-brane\,\,2d CFT with BH entropy
D3-brane\,\,N=4 D=4 super Yang-Mills theory
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection\,
string\,B2-field2d SCFT
NS5-brane\,B6-fieldlittle string theory
D-brane for topological string\,
M-brane11D SuGra/M-theorycircle n-connection\,
M2-brane\,C3-fieldABJM theory, BLG model
M5-brane\,C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane\,C6-field on G2-manifold
membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d


The original article on the N=6N=6-case is

and for discrete torsion in the supergravity C-field in

inspired by the N=8N=8-case of the BLG model

with precursor considerations in

The N=5N=5-case is discussed in

The N=4N=4-case is discussed in

  • Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, Jaemo Park, N=4 Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets, JHEP 0807:091,2008 (arXiv:0805.3662)

  • Fa-Min Chen, Yong-Shi Wu, Superspace Formulation in a Three-Algebra Approach to D=3, N=4,5 Superconformal Chern-Simons Matter Theories, Phys.Rev.D82:106012, 2010 (arXiv:1007.5157)

More on the role of discrete torsion in the supergravity C-field is in

Discussion of boundary conditions leading to brane intersection laws with the M-wave, black M5-brane and MO9 is in

Review includes

Discussion of extension to boundary field theory (describing M2-branes ending on M5-branes) includes

A kind of double dimensional reduction of the ABJM model to something related to type II superstrings and D1-branes is discussed in

Discussion of the ABJM model in Horava-Witten theory and reducing to heterotic strings is in

Discussion of the model as a higher gauge theory (due to its coupling to the supergravity C-field) is in

Classification of the possible superpotentials? via representation theory is due to

and derived from this a classification of the possible orbifolding (see at spherical space form: 7d with spin structure) is in

See also

  • Nadav Drukker, Marcos Marino, Pavel Putrov, From weak to strong coupling in ABJM theory (arXiv:1007.3837)

  • Shai M. Chester, Silviu S. Pufu, Xi Yin, The M-Theory S-Matrix From ABJM: Beyond 11D Supergravity (arXiv:1804.00949)

Computation of black hole entropy in 4d via AdS4-CFT3 duality from holographic entanglement entropy in the ABJM theory for the M2-brane is discussed in

  • Jun Nian, Xinyu Zhang, Entanglement Entropy of ABJM Theory and Entropy of Topological Black Hole (arXiv:1705.01896)

Last revised on May 22, 2019 at 15:15:07. See the history of this page for a list of all contributions to it.