For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
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$(2k+1/4)\phantom{A}$ | $\phantom{A}SO(2k+1)\phantom{A}$ | |
$\phantom{A}3\phantom{A}$ | $\phantom{A}2k\phantom{A}$ | $\phantom{A}D(k,2)\simeq$ 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 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) |
Under holographic duality supposed to be related to M-theory on $AdS_4 \times S^7 / \mathbb{Z}_k$.
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).
The original article on the $N=6$-case is
inspired by the $N=8$-case of the BLG model
Jonathan Bagger, Neil Lambert, Modeling Multiple M2’s, Phys. Rev. D75, 045020 (2007). (hep-th/0611108).
Jonathan Bagger, Neil Lambert, Phys. Rev. D77, 065008 (2008). (arXiv:0711.0955).
The $N=5$-case is discussed in
Kazuo Hosomichi, Ki-Myeong Lee, Sangmin Lee, Sungjay Lee, Jaemo Park, $N=5,6$ 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 Louis 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, 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)
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)
Review includes
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)
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)
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
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
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, Euclid)
José Figueroa-O'Farrill, M2-branes, ADE and Lie superalgebras, talk at IPMU 2009 (pdf)
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
Last revised on May 2, 2018 at 12:18:14. See the history of this page for a list of all contributions to it.