nLab
3d superconformal gauge field theory

Context

Quantum field theory

Super-Geometry

String theory

Contents

Idea

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

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+1/4)\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/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/k+1)\phantom{A}AU(k+1)A\phantom{A}U(k+1)\phantom{A}D3-brane
4d superconformal gauge field theory
A5A\phantom{A}5\phantom{A}A1A\phantom{A}1\phantom{A}AF(4)A\phantom{A}F(4)\phantom{A}ASO(3)A\phantom{A}SO(3)\phantom{A}
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(8/2k)\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)

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:

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)

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

References

The original article on the N=8N=8-case (the BLG model):

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

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)

Review includes

For more see at ABJM model.

Last revised on April 25, 2018 at 11:44:18. See the history of this page for a list of all contributions to it.