nLab D=4 N=1 super Yang-Mills theory

Idea

super Yang-Mills theory on a 4-dimensional spacetime with $N = 1$ supersymmetry.

$d$	$N$	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)

Related concepts

On ordinary Yang-Mills theory (YM):

• D=2 YM, D=3 YM, D=4 YM, D=5 YM, D=6 YM, D=7 YM, D=8 YM

• Maxwell theory/electromagnetism (U(1) YM), Donaldson theory (SU(2) YM), quantum chromodynamics (SU(3) YM)

• Yang-Mills equation, linearized Yang-Mills equation, Yang-Mills instanton, Yang-Mills field, stable Yang-Mills connection, Yang-Mills moduli space, Yang-Mills flow, F-Yang-Mills equation, Bi-Yang-Mills equation

• Uhlenbeck's singularity theorem, Uhlenbeck's compactness theorem

On variants of Yang-Mills theory and on super Yang-Mills theory (SYM):

• Yang-Mills-Higgs equations, stable Yang-Mills-Higgs pair, Yang-Mills-Higgs flow

• Einstein-Yang-Mills theory, Einstein-Yang-Mills-Dirac theory, Einstein-Yang-Mills-Dirac-Higgs theory

• 3D Yang-Mills-Chern-Simons theory

• 3d superconformal gauge field theory: D=3 N=1 SYM, D=3 N=2 SYM, D=3 N=4 SYM

• 4d superconformal gauge field theory: D=4 N=1 SYM, D=4 N=2 SYM, D=4 N=4 SYM

• D=5 SYM, D=6 SYM, D=7 SYM, D=10 SYM

• topological Yang-Mills theory, topologically twisted D=4 super Yang-Mills theory

• geometric engineering of quantum field theory

$d$	$N$	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

General

Original articles:

• Hugh Osborn, $\mathcal{N}=1$ Superconformal Symmetry in Four Dimensional Quantum Field Theory, Annals Phys. 272:243-294, 1999 (arXiv:hep-th/9808041)

Review:

• Yuji Tachikawa, Lectures on $4d$ $N=1$ dynamics and related topics (arXiv:1812.08946)

See also at N=2 D=4 super Yang-Mills theory.

The KK-compactification of the D=6 N=(1,0) SCFT (on M5-branes) to D=4 N=1 super Yang-Mills:

• Ibrahima Bah, Christopher Beem, Nikolay Bobev, Brian Wecht, Four-Dimensional SCFTs from M5-Branes (arXiv:1203.0303)

• Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir, $4d$ $\mathcal{N} = 1$ from $6d (1,0)$, J. High Energ. Phys. (2017) 2017: 64 (arXiv:1610.09178)

• Ibrahima Bah, Amihay Hanany, Kazunobu Maruyoshi, Shlomo S. Razamat, Yuji Tachikawa, Gabi Zafrir, $4d$ $\mathcal{N}=1$ from $6d$ $\mathcal{N}=(1,0)$ on a torus with fluxes (arXiv:1702.04740)

• Hee-Cheol Kim, Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir, E-String Theory on Riemann Surfaces, Fortsch. Phys. (arXiv:1709.02496)

• Hee-Cheol Kim, Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir, D-type Conformal Matter and SU/USp Quivers, JHEP06(2018)058 (arXiv:1802.00620)

• Hee-Cheol Kim, Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir, Compactifications of ADE conformal matter on a torus, JHEP09(2018)110 (arXiv:1806.07620)

• Shlomo S. Razamat, Gabi Zafrir, Compactification of 6d minimal SCFTs on Riemann surfaces, Phys. Rev. D 98, 066006 (2018) (arXiv:1806.09196)

• Jin Chen, Babak Haghighat, Shuwei Liu, Marcus Sperling, 4d N=1 from 6d D-type N=(1,0) (arXiv:1907.00536)

Via D3 branes at resolved orbifold singularities and invoking a generalized McKay correspondence:

• Pietro G. Fré, Lectures on resolutions à la Kronheimer of orbifold singularities, McKay quivers for Gauge Theories on D3 branes, and the issue of Ricci flat metrics on the resolved three-folds [arXiv:2308.14022]

The Witten index:

• Leonardo Rastelli, Shlomo S. Razamat, The supersymmetric index in four dimensions, Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 44 (arXiv:1608.02965)