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

• D=4 Yang-Mills theory

• quiver gauge theory, fractional D-brane

• super Yang-Mills theory

  • N=4 D=4 super Yang-Mills theory

  • N=4 D=3 super Yang-Mills theory

  • topologically twisted D=4 super Yang-Mills theory

    • 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)