# nLab ABJM theory

Contents

### Context

#### $\infty$-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## Idea

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

$d$$N$superconformal super Lie algebraR-symmetryblack brane worldvolume
superconformal field theory
$\phantom{A}3\phantom{A}$$\phantom{A}2k+1\phantom{A}$$\phantom{A}B(k,2) \simeq$ osp$(2k+1/4)\phantom{A}$$\phantom{A}SO(2k+1)\phantom{A}$
$\phantom{A}3\phantom{A}$$\phantom{A}2k\phantom{A}$$\phantom{A}D(k,2)\simeq$ osp$(2k/4)\phantom{A}$$\phantom{A}SO(2k)\phantom{A}$M2-brane
3d superconformal gauge field theory
$\phantom{A}4\phantom{A}$$\phantom{A}k+1\phantom{A}$$\phantom{A}A(3,k)\simeq \mathfrak{sl}(4/k+1)\phantom{A}$$\phantom{A}U(k+1)\phantom{A}$D3-brane
4d superconformal gauge field theory
$\phantom{A}5\phantom{A}$$\phantom{A}1\phantom{A}$$\phantom{A}F(4)\phantom{A}$$\phantom{A}SO(3)\phantom{A}$
$\phantom{A}6\phantom{A}$$\phantom{A}k\phantom{A}$$\phantom{A}D(4,k) \simeq$ osp$(8/2k)\phantom{A}$$\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 $k = 2$ the supersymmetry of the ABJM model increases to $\mathcal{N} = 8$. For $k = 2$ and $N = 2$ 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/$\mathbb{Z}/k$ (see also MFFGME 09).

Notice that the worldvolume $SU(N)$ gauge group enhancement at an $\mathbb{Z}_k$-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) $\geq \tfrac{1}{2}$ BPS black M2-brane-solutions of the equations of motion of 11-dimensional supergravity shows that these are the Cartesian product $AdS_4 \times (S^7/G)$ of 4-dimensional anti de Sitter spacetime with a 7-dimensional spherical space form $S^7/{\widehat{G}}$ with spin structure and $N \geq 4$, for $\widehat{G}$ a finite subgroup of SU(2) (MFFGME 09, see here).

$N$ Killing spinors on
spherical space form $S^7/\widehat{G}$
$\phantom{AA}\widehat{G} =$spin-lift of subgroup of
isometry group of 7-sphere
3d superconformal gauge field theory
on back M2-branes
with near horizon geometry $AdS_4 \times S^7/\widehat{G}$
$\phantom{AA}N = 8\phantom{AA}$$\phantom{AA}\mathbb{Z}_2$cyclic group of order 2BLG model
$\phantom{AA}N = 7\phantom{AA}$
$\phantom{AA}N = 6\phantom{AA}$$\phantom{AA}\mathbb{Z}_{k\gt 2}$cyclic groupABJM model
$\phantom{AA}N = 5\phantom{AA}$$\phantom{AA}2 D_{k+2}$
$2 T$, $2 O$, $2 I$
binary dihedral group,
binary tetrahedral group,
binary octahedral group,
binary icosahedral group
$\phantom{AA}N = 4\phantom{AA}$$\phantom{A}2 D_{k+2}$
$2 O$, $2 I$
binary dihedral group,
binary octahedral group,
binary icosahedral group
(HLLLP 08b, Chen-Wu 10)

## Properties

Under holographic duality supposed to be related to M-theory on $AdS_4 \times S^7 / \mathbb{Z}_k$.

### 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
$(D = 2n)$type IIA$\,$$\,$
D(-2)-brane$\,$$\,$
D0-brane$\,$$\,$BFSS matrix model
D2-brane$\,$$\,$$\,$
D4-brane$\,$$\,$D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane$\,$$\,$D=7 super Yang-Mills theory
D8-brane$\,$$\,$
$(D = 2n+1)$type IIB$\,$$\,$
D(-1)-brane$\,$$\,$$\,$
D1-brane$\,$$\,$2d CFT with BH entropy
D3-brane$\,$$\,$N=4 D=4 super Yang-Mills theory
D5-brane$\,$$\,$$\,$
D7-brane$\,$$\,$$\,$
D9-brane$\,$$\,$$\,$
(p,q)-string$\,$$\,$$\,$
(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$\,$
A-brane$\,$
B-brane$\,$
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
M-wave
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane$\,$C6-field on G2-manifold
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

## References

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

and for discrete torsion in the supergravity C-field in

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

The $N=5$-case is discussed in

The $N=4$-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