nLab ABJM theory

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 algebra	R-symmetry	black brane worldvolume superconformal field theory via AdS-CFT
$\phantom{A}3\phantom{A}$	$\phantom{A}2k+1\phantom{A}$	$\phantom{A}B(k,2) \simeq$ osp$(2k+1 \vert 4)\phantom{A}$	$\phantom{A}SO(2k+1)\phantom{A}$
$\phantom{A}3\phantom{A}$	$\phantom{A}2k\phantom{A}$	$\phantom{A}D(k,2)\simeq$ osp$(2k \vert 4)\phantom{A}$	$\phantom{A}SO(2k)\phantom{A}$	M2-brane D=3 SYM BLG model ABJM model
$\phantom{A}4\phantom{A}$	$\phantom{A}k+1\phantom{A}$	$\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}$	$\phantom{A}U(k+1)\phantom{A}$	D3-brane D=4 N=4 SYM D=4 N=2 SYM D=4 N=1 SYM
$\phantom{A}5\phantom{A}$	$\phantom{A}1\phantom{A}$	$\phantom{A}F(4)\phantom{A}$	$\phantom{A}SO(3)\phantom{A}$	D4-brane D=5 SYM
$\phantom{A}6\phantom{A}$	$\phantom{A}k\phantom{A}$	$\phantom{A}D(4,k) \simeq$ osp$(8 \vert 2k)\phantom{A}$	$\phantom{A}Sp(k)\phantom{A}$	M5-brane D=6 N=(2,0) SCFT D=6 N=(1,0) SCFT

(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 2	BLG model
$\phantom{AA}N = 7\phantom{AA}$	—	—	—
$\phantom{AA}N = 6\phantom{AA}$	$\phantom{AA}\mathbb{Z}_{k\gt 2}$	cyclic group	ABJM 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	(HLLLP 08a, BHRSS 08)
$\phantom{AA}N = 4\phantom{AA}$	$\phantom{A}2 D_{k+2}$ $2 O$, $2 I$	binary dihedral group, binary octahedral group, binary icosahedral group	(HLLLP 08b, Chen-Wu 10)

• José Figueroa-O'Farrill et al. 2009 (arXiv:0909.0163, pdf slides)

Properties

AdS/CFT duality

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).

Related concepts

• super Chern-Simons theory

• BLG model

• membrane matrix model

$d$	$N$	superconformal super Lie algebra	R-symmetry	black brane worldvolume superconformal field theory via AdS-CFT
$\phantom{A}3\phantom{A}$	$\phantom{A}2k+1\phantom{A}$	$\phantom{A}B(k,2) \simeq$ osp$(2k+1 \vert 4)\phantom{A}$	$\phantom{A}SO(2k+1)\phantom{A}$
$\phantom{A}3\phantom{A}$	$\phantom{A}2k\phantom{A}$	$\phantom{A}D(k,2)\simeq$ osp$(2k \vert 4)\phantom{A}$	$\phantom{A}SO(2k)\phantom{A}$	M2-brane D=3 SYM BLG model ABJM model
$\phantom{A}4\phantom{A}$	$\phantom{A}k+1\phantom{A}$	$\phantom{A}A(3,k)\simeq \mathfrak{sl}(4 \vert k+1)\phantom{A}$	$\phantom{A}U(k+1)\phantom{A}$	D3-brane D=4 N=4 SYM D=4 N=2 SYM D=4 N=1 SYM
$\phantom{A}5\phantom{A}$	$\phantom{A}1\phantom{A}$	$\phantom{A}F(4)\phantom{A}$	$\phantom{A}SO(3)\phantom{A}$	D4-brane D=5 SYM
$\phantom{A}6\phantom{A}$	$\phantom{A}k\phantom{A}$	$\phantom{A}D(4,k) \simeq$ osp$(8 \vert 2k)\phantom{A}$	$\phantom{A}Sp(k)\phantom{A}$	M5-brane D=6 N=(2,0) SCFT D=6 N=(1,0) SCFT

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

References

Precursors

Precursor considerations in

• John Schwarz, Superconformal Chern-Simons Theories (arXiv:arXiv:hep-th/0411077

The lift of Dp-D(p+2)-brane bound states in string theory to M2-M5-brane bound states/E-strings in M-theory, under duality between M-theory and type IIA string theory+T-duality, via generalization of Nahm's equation (this eventually motivated the BLG-model/ABJM model):

• Anirban Basu, Jeffrey Harvey, The M2-M5 Brane System and a Generalized Nahm’s Equation, Nucl.Phys. B713 (2005) 136-150 (arXiv:hep-th/0412310)

• Jonathan Bagger, Neil Lambert, Sunil Mukhi, Constantinos Papageorgakis, Section 2.2.1 of Multiple Membranes in M-theory, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 (arXiv:1203.3546, doi:10.1016/j.physrep.2013.01.006)

This inspired the BLG model:

• Jonathan Bagger, Neil Lambert, Modeling Multiple M2’s, Phys. Rev. D75, 045020 (2007). (hep-th/0611108).

• Jonathan Bagger, Neil Lambert, Gauge Symmetry and Supersymmetry of Multiple M2-Branes, Phys. Rev. D77, 065008 (2008). (arXiv:0711.0955).

General

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

• Ofer Aharony, Oren Bergman, Daniel Jafferis, Juan Maldacena, $N=6$ superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 0810:091,2008, DOI:10.1088/1126-6708/2008/10/091 (arXiv:0806.1218)

and for discrete torsion in the supergravity C-field in

• Ofer Aharony, Oren Bergman, Daniel Jafferis, Fractional M2-branes, JHEP 0811:043, 2008 (arXiv:0807.4924)

  (on fractional M2-branes)

inspired by the $N=8$-case of the BLG model (Bagger-Lambert 06)

The $N=5$-case is discussed in

• Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, Jaemo Park, $\mathcal{N}=5,6$ Superconformal Chern-Simons Theories and M2-branes on Orbifolds, JHEP 0809:002, 2008 (arXiv:0806.4977)

• Eric Bergshoeff, Olaf Hohm, Diederik Roest, Henning Samtleben, Ergin Sezgin, The Superconformal Gaugings in Three Dimensions, JHEP0809:101, 2008 (arXiv:0807.2841)

• Ofer Aharony, Oren Bergman, Daniel Jafferis, Fractional M2-branes, JHEP 0811:043, 2008 (arXiv:0807.4924)

The $N=4$-case is discussed in

• Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, Jaemo Park, \mathcal{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

• Mauricio Romo, Aspects of ABJM orbifolds with discrete torsion, J. High Energ. Phys. (2011) 2011 (arXiv:1011.4733)

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

• Chong-Sun Chu, Douglas J. Smith, Multiple Self-Dual Strings on M5-Branes, JHEP 1001:001, 2010 (arXiv:0909.2333)

• David Berman, Malcolm J. Perry, Ergin Sezgin, Daniel C. Thompson, Boundary Conditions for Interacting Membranes, JHEP 1004:025, 2010 [arXiv:0912.3504, doi:10.1007/JHEP04(2010)025]

As a matrix model,:

• Asadig Mohammed, Jeff Murugan, Horatiu Nastase, Looking for a Matrix model of ABJM, Phys. Rev. D82:086004, 2010 (arXiv:1003.2599)

Review:

• Igor Klebanov, Giuseppe Torri, M2-branes and AdS/CFT, Int.J.Mod.Phys.A25:332-350, 2010 (arXiv;0909.1580)

• Neil B. Copland, Introductory Lectures on Multiple Membranes (arXiv:1012.0459)

• Neil Lambert, M-Theory and Maximally Supersymmetric Gauge Theories, Annual Review of Nuclear and Particle Science, Vol. 62:285-313 (arXiv:1203.4244, doi:10.1146/annurev-nucl-102010-130248)

• Jonathan Bagger, Neil Lambert, Sunil Mukhi, Constantinos Papageorgakis, Multiple Membranes in M-theory, Physics Reports, Volume 527, Issue 1, 2013 (arXiv:1203.3546, doi:10.1016/j.physrep.2013.01.006)

• Neil Lambert, Lessons from M2’s and Hopes for M5’s, Proceedings of the LMS-EPSRC Durham Symposium: Higher Structures in M-Theory 2018 Fortschritte der Physik, 2019 (arXiv:1903.02825, slides pdf, video recording)

• Manikantt Mummalaneni: AdS4/CFT3: ABJM Theory, Brane Geometry, Correlators and Mellin Space [arXiv:2408.11835]

Discussion of Montonen-Olive duality in D=4 super Yang-Mills theory via ABJM-model as D3-brane model after double dimensional reduction followed by T-duality:

• Koji Hashimoto, Ta-Sheng Tai, Seiji Terashima, Toward a Proof of Montonen-Olive Duality via Multiple M2-branes, JHEP 0904:025, 2009 (arxiv:0809.2137)

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

• David Berman, Daniel Thompson, Membranes with a boundary, Nucl.Phys.B820:503-533,2009 (arXiv:0904.0241)

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

• Horatiu Nastase, Constantinos Papageorgakis, Dimensional reduction of the ABJM model, JHEP 1103:094,2011 (arXiv:1010.3808)

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

• Neil Lambert, Heterotic M2-branes, Physics Letters B Volume 749, 7 October 2015, Pages 363-367 (arXiv:1507.07931)

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

• Sam Palmer, Christian Saemann, section 2 of M-brane Models from Non-Abelian Gerbes, JHEP 1207:010, 2012 (arXiv:1203.5757)

• Sam Palmer, Christian Saemann, The ABJM Model is a Higher Gauge Theory, IJGMMP 11 (2014) 1450075 (arXiv:1311.1997)

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

• Paul de Medeiros, José Figueroa-O'Farrill, Elena Méndez-Escobar, Superpotentials for superconformal Chern-Simons theories from representation theory, J. Phys. A 42:485204,2009 (arXiv:0908.2125)

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

• Paul de Medeiros, José Figueroa-O'Farrill, Sunil Gadhia, Elena Méndez-Escobar, Half-BPS quotients in M-theory: ADE with a twist, JHEP 0910:038,2009 (arXiv:0909.0163, pdf slides)

• Paul de Medeiros, José Figueroa-O'Farrill, Half-BPS M2-brane orbifolds, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (arXiv:1007.4761, eujclid:atmp/1408561553)

• José Figueroa-O'Farrill, M2-branes, ADE and Lie superalgebras, talk at IPMU 2009 (pdf)

Discussion via the conformal bootstrap:

• Nathan B. Agmon, Shai Chester, Silviu S. Pufu, The M-theory Archipelago (arXiv:1907.13222)

• Damon J. Binder, Shai Chester, Max Jerdee, Silviu S. Pufu, The 3d $\mathcal{N}=6$ Bootstrap: From Higher Spins to Strings to Membranes (arXiv:2011.05728)

See also

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

• Shai 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)

Discussion of higher curvature corrections in the abelian case:

• Shin Sasaki, On Non-linear Action for Gauged M2-brane, JHEP 1002:039, 2010 (arxiv:0912.0903)

On abelian anyons (or at least Aharonov-Bohm phases described holographically via the ABJM model:

• Shoichi Kawamoto, Feng-Li Lin, Holographic Anyons in the ABJM Theory, JHEP 1002:059 (2010) [arXiv:0910.5536]

Mass deformation

The Myers effect in M-theory for M2-branes polarizing into M5-branes of (fuzzy) 3-sphere-shape (M2-M5 brane bound states):

• Iosif Bena, The M-theory dual of a 3 dimensional theory with reduced supersymmetry, Phys. Rev. D62:126006, 2000 (arXiv:hep-th/0004142)

• Masato Arai, Claus Montonen, Shin Sasaki, Vortices, Q-balls and Domain Walls on Dielectric M2-branes, JHEP 0903:119, 2009 (arXiv:0812.4437)

• Iosif Bena, Mariana Graña, Stanislav Kuperstein, Stefano Massai, Tachyonic Anti-M2 Branes, JHEP 1406:173, 2014 (arXiv:1402.2294)

With emphasis on the role of the Page charge/Hopf WZ term:

• Krzysztof Pilch, Alexander Tyukov, Nicholas Warner, Flowing to Higher Dimensions: A New Strongly-Coupled Phase on M2 Branes, JHEP11 (2015) 170 (arXiv:1506.01045)

Via the mass-deformed ABJM model:

• Jaume Gomis, Diego Rodriguez-Gomez, Mark Van Raamsdonk, Herman Verlinde, A Massive Study of M2-brane Proposals, JHEP 0809:113, 2008 (arXiv:0807.1074)

• Jonathan Bagger, Neil Lambert, Sunil Mukhi, Constantinos Papageorgakis, Section 6.4 of: Multiple Membranes in M-theory, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 (arXiv:1203.3546, doi:10.1016/j.physrep.2013.01.006)

The corresponding D2-NS5 bound state under duality between M-theory and type IIA string theory:

• Iosif Bena, Aleksey Nudelman, Warping and vacua of $(S)YM_{3+1}$, Phys. Rev. D62 (2000) 086008 (arXiv:hep-th/0005163)

On Wilson line-quantum observables and bosonization in the ABJM model:

• Amit Sever, Line Operators in Chern-Simons-Matter Theories and Bosonization in Three Dimensions, talk at Strings 2022 [indico:4940843, slides, video]