nLab 3d superconformal gauge field theory

Idea

According to the classification of superconformal symmetry, there should exists superconformal field theories in 3 dimensions

$d$	$N$	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)

Such superconformal gauge field theories involving Chern-Simons theory coupled to matter exist and are thought to describe the worldvolume theory of black M2-branes located at ADE-singularities:

$N$ 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)

The corresponding story for the M5-brane is the 6d superconformal gauge field theory.

Related concepts

• D=3 N=2 super Yang-Mills theory

• D=3 N=4 super Yang-Mills theory

References

The original article on the $N=8$-case (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 original article on the $N=6$-case (the ABJM model):

• Ofer Aharony, Oren Bergman, Daniel Louis 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)

The $\mathcal{N}=5$-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 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, $\mathcal{N}=4$ 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)

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)

For more see at ABJM model.