nLab D=4 N=2 super Yang-Mills theory

Idea

The special case of super Yang-Mills theory over a spacetime of dimension 4 and with $N = 2$ 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)

Properties

Moduli space of vacua

A speciality of $N=2$, $D = 4$ SYM is that its moduli space of vacua has two “branches” called the Coulomb branch and the Higgs branch. This is the content of what is now called Seiberg-Witten theory (Seiberg-Witten 94). Review includes (Albertsson 03, section 2.3.4).

Confinement

While confinement in plain Yang-Mills theory is still waiting for mathematical formalization and proof (see Jaffe-Witten), N=2 D=4 super Yang-Mills theory is has been obsrved in (Seiberg-Witten 94).

Reduction to $D = 3$ super Yang-Mills

By dimensional reduction on $\mathbb{R}^3 \times S^1$ families of $N = 2, D = 4$ SYM theories interpolate to N=4 D=3 super Yang-Mills theory. (Seiberg-Witten 96).

Construction by compactification of 5-branes

$N=2$ super Yang-Mills theory can be realized as the worldvolume theory of M5-branes compactified on a Riemann surface (Klemm-Lerche-Mayr-Vafa-Warner 96, Witten 97, Gaiotto 09), hence as a compactifiction of the 6d (2,0)-superconformal QFT on the M5. This in particular gives a geometric interpretation of Seiberg-Witten duality in 4d in terms of the 6d 5-brane geometry.

Specifically the AGT correspondence expresses this relation in terms of the partition function of the theory and a 2d CFT on the Riemann surface on which the 5-brane is compactified. See at AGT correspondence for more on this.

gauge theory induced via AdS-CFT correspondence

M-theory perspective via AdS7-CFT6	F-theory perspective
11d supergravity/M-theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$	compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
$\;\;\;\;\downarrow$ topological sector
7-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invariance	M5-brane worldvolume theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface	double dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondence	D3-brane worldvolume theory with type IIB S-duality
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$, Donaldson theory

$\,$

gauge theory induced via AdS5-CFT4
type II string theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$
$\;\;\;\; \downarrow$ topological sector
5-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence

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

• quiver gauge theory, fractional D-brane

• chiral ring, quantum cohomology

• Seiberg-Witten theory, electric-magnetic duality, symplectic duality

• AGT correspondence

• spectral network

• 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

Introductions and surveys

• Davide Gaiotto, Recent progress in $N=2$ $4d$ field theory (2009) (pdf)

• Gregory Moore, Four-dimensional $N=2$ Field Theory and Physical Mathematics (arXiv:1211.2331)

• Greg Moore, Surface Defects and the BPS Spectrum of $4d$ $N=2$ Theories (pdf)

• Yuji Tachikawa, $\mathcal{N}=2$ supersymmetric dynamics for pedestrians, Lecture Notes in Physics 890 (2014) [arXiv:1312.2684, doi:10.1007/978-3-319-08822-8, also “…for dummies”: webpage, pdf]

• Mohammad Akhond, Guillermo Arias-Tamargo, Alessandro Mininno, Hao-Yu Sun, Zhengdi Sun, Yifan Wang, Fengjun Xu, The Hitchhiker’s Guide to 4d $\mathcal{N}=2$ Superconformal Field Theories [arXiv:2112.14764]

General

The terminology “Coulomb branch” and “Higgs branch” first appears in

• Nathan Seiberg, Edward Witten, Monopoles, Duality and Chiral Symmetry Breaking in $N=2$ Supersymmetric QCD (arXiv:hep-th/9408099)

See also at Seiberg-Witten theory:

The dimensional reduction to $D = 3$ was first considered in

• Nathan Seiberg, Edward Witten, Gauge Dynamics And Compactification To Three Dimensions (arXiv:hep-th/9607163)

The confinement-phenomenon was observed in

• Nathan Seiberg, Edward Witten, Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory, Nucl.Phys.B426:19-52,1994; Erratum-ibid.B430:485-486,1994 (arXiv:hep-th/9407087)

Reviews of this confinement mechanism:

• Alexei Yung, What Do We Learn about Confinement from the Seiberg-Witten Theory (arXiv:hep-th/0005088)

• Cecilia Albertsson, Superconformal D-branes and moduli spaces (arXiv:hep-th/0305188)

• Yang-Hui He, Lectures on D-branes, Gauge Theories and Calabi-Yau Singularities, International Summer School in Mathematical Physics (arXiv:hep-th/0408142)

• Emily Nardoni, From $\mathcal{N} = 2$ Supersymmetry to adjoint QCD, talk at Strings 2022 [indico:4940894, pdf, video]

For references on wall crossing of BPS states see the references given there.

Construction from 5-branes

$N=2$ $D=4$ SYM including its Seiberg-Witten theory (Seiberg-Witten 94) may be understood as being the compactification of the 6d (2,0)-superconformal QFT on the worldvolume of M5-branes on a Riemann surface: the Riemann surface is identified with the Seiberg-Witten curve of complexified coupling constants. This observation goes back to

• Albrecht Klemm, Wolfgang Lerche, Peter Mayr, Cumrun Vafa, Nicholas Warner, Self-Dual Strings and $\mathcal{N}=2$ Supersymmetric Field Theory, Nucl. Phys. B 477 (1996) 746-766 [arXiv:hep-th/9604034, doi:10.1016/0550-3213(96)00353-7]

• Edward Witten, Solutions Of Four-Dimensional Field Theories Via M Theory, Nucl. Phys. B 500 (1997) 3-42 [arXiv:hep-th/9703166, doi:10.1016/S0550-3213(97)00416-1]

review in:

• Csaba Csaki, Joshua Erlich, John Terning, pp. 4 of; Seiberg-Witten Description of the Deconstructed 6D $(0,2)$ Theory, Phys. Rev. D 67 025019 (2003) [arXiv:hep-th/0208095, doi:10.1103/PhysRevD.67.025019]

• Ling Bao, Elli Pomoni, Masato Taki, Futoshi Yagi, p. 5 of: M5-branes, toric diagrams and gauge theory duality, J. High Energ. Phys. 2012 105 (2012) [arXiv:1112.5228, doi:10.1007/JHEP04(2012)105]

• Taro Kimura, §4.5.2 in: Seiberg–Witten Geometry, Chapter 4 in: Instanton Counting, Quantum Geometry and Algebra, Mathematical Physics Studies, Springer (2021) [doi:10.1007/978-3-030-76190-5_4]

The further observation that therefore the sewing of Riemann surfaces on which one compactifies the M5-brane yields a gluing operation on N=2 SYM theories is due to

• Davide Gaiotto, $N=2$ dualities, JHEP 08 (2012) 034 [arXiv:0904.2715]

cf. at AGT correspondence.

The topological twisting of the compactification which is used around (2.27) there was previously introduced in section 3.1.2 of:

• Davide Gaiotto, Gregory Moore Andrew Neitzke, Wall-crossing, Hitchin Systems, and the WKB Approximation, (arXiv:0907.3987)

and is discussed also for instance in section 5.1 of

• Edward Witten, Fivebranes and knots (arXiv:1101.3216)

(This is possibly also the mechanism behind the AGT correspondence, though the details behind that statement seem to be unclear.)

A brief review of these matters is in (Moore 12, section 7). A formalization of the topological twist in perturbation theory formalized by factorization algebras with values in BV complexes is in section 16 of

• Kevin Costello, Notes on supersymmetric and holomorphic field theories in dimension 2 and 4, talk at Geometric and Algebraic Structures in Mathematics, Stony Brook (2011) [arXiv:1111.4234]

For more on this see at topologically twisted D=4 super Yang-Mills theory.

An amplification of the relevance of this to the understanding of S-duality/electric-magnetic duality is in

• Edward Witten, Conformal field theory in four and six dimensions, in Ulrike Tillmann (ed.) Topology, geometry and quantum field theory LMS Lecture Note Series (2004) (arXiv:0712.0157)

and the resulting relation to the geometric Langlands correspondence is discussed in

• Edward Witten, Geometric Langlands From Six Dimensions (arXiv:0905.2720)

  .

The corresponding dual theory under AdS-CFT duality is discussed in

• Davide Gaiotto, Juan Maldacena, The gravity duals of N=2 superconformal field theories (arXiv:0904.4466)

Discussion of construction of just N=1 D=4 super Yang-Mills theory this way is in

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

Relation to 3-branes in M5-branes (M5 self-intersections)

Relation to the 3-brane in 6d:

• Paul Howe, Neil Lambert, Peter West, Classical M-Fivebrane Dynamics and Quantum $N=2$ Yang-Mills, Phys. Lett. B418 (1998) 85-90 (arXiv:hep-th/9710034)

• Neil Lambert, Peter West, Gauge Fields and M-Fivebrane Dynamics, Nucl. Phys. B524 (1998) 141-158 (arXiv:hep-th/9712040)

• Neil Lambert, Peter West, $N=2$ Superfields and the M-Fivebrane, Phys. Lett. B424 (1998) 281-287 (arXiv:hep-th/9801104)

• Neil Lambert, Peter West, Monopole Dynamics from the M-Fivebrane, Nucl. Phys. B556 (1999) 177-196 (arXiv:hep-th/9811025)

and via F-theory in

• Robert de Mello Koch, Alastair Paulin-Campbell, Joao P. Rodrigues, Monopole Dynamics in $\mathcal{N}=2$ super Yang-Mills Theory From a Threebrane Probe, Nucl. Phys. B559 (1999) 143-164 (arXiv:hep-th/9903207)

Dual superconductor model of confinement

General

The original proposal of the dual superconductor model of color confinement:

• Stanley Mandelstam: Vortices and quark confinement in non-abelian gauge theories, Physics Letters B 53 5 (1975) 476–478 [doi:10.1016/0370-2693(75)90221-X]

• Gerard ’t Hooft; p 4-5 in: Gauge Fields with Unified Weak, Electromagnetic, and Strong Interactions, Rapporteur’s talk at E.P.S. International Conference on High Energy Physics, Palermo, Sicily (June 1975) [spire:2781, pdf, pdf]

• Gerard ’t Hooft: On the phase transition towards permanent quark confinement, Nuclear Physics B 138 1 (1978) 1–25 [doi:10.1016/0550-3213(78)90153-0]

Further discussion:

• L. Del Debbio, M. Faber, Jeff Greensite, S. Olejnik: Some Cautionary Remarks on Abelian Projection and Abelian Dominance, Nucl. Phys. Proc. Suppl. 53 (1997) 141–147 [doi:10.1016/S0920-5632(96)00608-1, arXiv:hep-lat/9607053]

• Y. M. Cho: Dimensional Transmutation by Monopole Condensation in QCD, Phys. Rev. D. 87 (2013) 085025 [doi:10.1103/PhysRevD.87.085025, arXiv:1206.6936]

Review:

• Thomas Schaefer, Edward Shuryak, section III D of: Instantons in QCD, Rev. Mod. Phys. 70 (1998) 323–426 [doi:10.1103/RevModPhys.70.323, arXiv:hep-ph/9610451]

• Adriano Di Giacomo: Confinement of Color by Dual Superconductivity, Acta Physica Polonica B 28 12 (1997)

• Jeff Greensite: The Confinement Problem in Lattice Gauge Theory, Prog. Part. Nucl. Phys. 51 (2003) 1 [doi:10.1016/S0146-6410(03)90012-3, arXiv:hep-lat/0301023]

• Adriano Di Giacomo: Confinement of Color: Recent Progress [arXiv:hep-lat/0310021]

• Georges Ripka: Dual superconductor models of color confinement, Lecture Notes in Physics 639, Springer (2004) [doi:10.1007/b94800, arXiv:hep-ph/0310102]

• Adriano Di Giacomo: A Strategy to Study Confinement in QCD, Braz. J. Phys. 37 (2007) 208–213 [arXiv:hep-lat/0610027]

• Adriano Di Giacomo: Confinement by dual superconductivity: an update (2001) [pdf]

• Adriano Di Giacomo: The Dual Superconductor Picture for Confinement, in: Confinement, Duality, and Non-Perturbative Aspects of QCD, NATO Science Series: B 368, Springer (2002) 415–437 [doi:10.1007/0-306-47056-X_15]

• Maxim Chernodub: QCD Vacuum as Dual Superconductor: Quark Confinement and Topology, Handbook of Nuclear Physics, Springer (2023) [doi:10.1007/978-981-19-6345-2_23]

In lattice gauge theory

On the dual superconductor model of confinement in view of lattice QCD computations/simulations:

• Michael E. Peskin: Mandelstam ‘t Hooft Duality in Abelian Lattice Models, Annals Phys. 113 (1978) 122 [doi:10.1016/0003-4916(78)90252-X]

• Paolo Cea, Leonardo Cosmai: Lattice investigation of dual superconductor mechanism of confinement, Nuclear Physics B - Proceedings Supplements 30 (1993) 572–575 [doi:10.1016/0920-5632(93)90276-C]

• Paolo Cea, Leonardo Cosmai: Dual Superconductor Mechanism of Confinement on the Lattice, Nuov Cim A 107 (1994) 541547 [doi:10.1007/BF02768788, arXiv:hep-lat/9210030]

• Paolo Cea, Leonardo Cosmai: The $SU(2)$ Confining Vacuum as a Dual Superconductor, Nucl. Phys. Proc. Suppl. 47 (1996) 318–321 [doi:10.1016/0920-5632(96)00065-5, arXiv:hep-lat/9509007]

• Hideo Suganuma, Naoyuki Sakumichi: Perfect Abelian dominance of confinement in quark-antiquark potential in $SU(3)$ lattice QCD [arXiv:1412.8489]

• Hideo Suganuma, Naoyuki Sakumichi: The three-quark potential and perfect Abelian dominance in $SU(3)$ lattice QCD, PoS LATTICE2015 (2016) 323 [arXiv:1511.05244 hep-lat]

• Paolo Cea, Leonardo Cosmai, Francesca Cuteri, Alessandro Papa: Flux tubes in the QCD vacuum, Phys. Rev. D 95 (2017) 114511 [doi:10.1103/PhysRevD.95.114511, arXiv:1702.06437 hep-lat]

• Hideo Suganuma, Naoyuki Sakumichi: Monopole Dominance of Confinement in $SU(3)$ Lattice QCD, PoS 336 267 (2018) [arXiv:1812.06827 hep-lat, pos:336/267, pdf]

• Zeinab Dehghan, Manfried Faber: What do we know about the confinement mechanism?, Phys. Part. Nuclei 56 (2025) 1148–1154 [doi:10.1134/S1063779625700236, arXiv:2412.10767 hep-lat]

In super-Yang-Mills theory

The dual superconductor model of confinement becomes analytically exact in $\mathcal{N}-2$ $D=4$ super Yang-Mills theory (cf. Seiberg-Witten theory) when broken to $\mathcal{N}=1$:

• Nathan Seiberg, Edward Witten: Monopole Condensation, And Confinement in $\mathcal{N}=2$ Supersymmetric Yang-Mills Theory, Nucl. Phys. B 426 (1994) 19–52 Erratum-ibid. B 430 (1994) 485-0486 [doi:10.1016/0550-3213(94)90124-4, arXiv:hep-th/9407087]

• Nathan Seiberg, Edward Witten: Monopoles, Duality and Chiral Symmetry Breaking in $\mathcal{N}=2$ Supersymmetric QCD, Nucl. Phys. B 431 (1994) 484–550 [doi:10.1016/0550-3213(94)90214-3, arXiv:hep-th/9408099]

Reviews with discussion of the impact on confinement in plain YM:

• Alexei Yung: What Do We Learn about Confinement from the Seiberg-Witten Theory, 3rd Moscow School of Physics and 28th ITEP Winter School of Physics [spire:527017,arXiv:hep-th/0005088]

• Michael Dine: On the Possibility of Demonstrating Confinement in Non-Supersymmetric Theories by Deforming Confining Supersymmetric Theories [arXiv:2211.17134]

In solid state physics

On experimentally accessible analogs of the dual superconductor model of confinement in solid state physics:

• M. Cristina Diamantini, Carlo A. Trugenberger, Valeri M. Vinokur: Confinement and Asymptotic Freedom with Cooper pairs, Nature Comm. Phys. 1 77 (2018) [doi:10.1038/s42005-018-0073-9, arXiv:1807.01984]

• M. Cristina Diamantini, Carlo A. Trugenberger: Superinsulators: a toy realization of QCD in condensed matter, Ch. 23 in Roman Jackiw – 80th Birthday Festschrift, World Scientific (2020) 275-286 [arXiv:2008.12541]

Review:

• Carlo A. Trugenberger: Superinsulators, Bose Metals and High-$T_c$ Superconductors: The Quantum Physics of Emergent Magnetic Monopoles, World Scientific (2022) [doi:10.1142/12688]

• Carlo A. Trugenberger: Superinsulation: Magnetic Monopoles and Electric Confinement in Condensed Matter, talk at CQTS (Feb 2025) [slides:pdf]