nLab SCFT

Idea

CFT with supersymmetry, i.e. with superconformal symmetry

$d$	$N$	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)

Examples

$d = 2$

• 2d SCFT

  • NSR string

  • heterotic string, 2d (2,0)-superconformal QFT

$d = 3$

• 3d superconformal gauge field theory

$d = 4$

• N=4 D=4 super Yang-Mills theory

$d = 6$

• D=6 N=(2,0) SCFT

• D=6 N=(1,0) SCFT

Applications

• mirror symmetry

Related concepts

• superconformal geometry

• supersymmetric sigma-model

• AdS/CFT correspondence

References

General

Survey of SCFTs in higher dimensions (D=4 SYM, D=5 SYM, D=6 N=1, D=6, N=2) their moduli (Higgs branches/Coulomb branches) and their dualities (KK-compactification, AGT correspondence, AdS/CFT duality, …):

• Shlomo Razamat, News from the higher $\mathcal{D}$ SQFT frontier, talk at Strings 2022 [indico:4940856, slides, video]

Classification

• Werner Nahm, Supersymmetries and their Representations, Nucl.Phys. B135 (1978) 149 (spire, pdf)

• Steven Shnider, The superconformal algebra in higher dimensions, Letters in Mathematical Physics November 1988, Volume 16, Issue 4, pp 377-383

• Shiraz Minwalla, Restrictions imposed by superconformal invariance on quantum field theories Adv. Theor. Math. Phys. 2, 781 (1998)

  (arXiv:hep-th/9712074).

• Clay Cordova, Thomas Dumitrescu, Kenneth Intriligator, Deformations of Superconformal Theories, (arXiv:1602.01217)