nLab Seiberg-Witten theory

Idea

Seiberg-Witten theory studies the moduli space of vacua in N=2 D=4 super Yang-Mills theory, in particular the electric-magnetic duality (Montonen-Olive duality) of the theory.

Related concepts

• N=2 D=4 super Yang-Mills theory

• Coulomb branch, Higgs branch

• Donaldson theory

• Chern-Simons theory, topological Yang-Mills theory

• topologically twisted D=4 super Yang-Mills theory

• AGT correspondence

References

General

For more and for general references see at N=2 D=4 super Yang-Mills theory.

The original articles:

• Nathan Seiberg, Edward Witten, Monopole condensation, and confinement in $N=2$ supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19-52;

  Erratum-ibid. B 430 (1994) 485-486, (arXiv:hep-th/9407087)

• Nathan Seiberg, Edward Witten, Monopoles, duality and chiral symmetry breaking in $N=2$

  supersymmetric QCD, Nucl. Phys. B431 (1994) 484-550 [hep-th/9408099]

Review:

• Taro Kimura, 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]

• Jürgen Einhorn, Thomas Friedrich, Seiberg-Witten theory (pdf)

• Simon Donaldson, The Seiberg-Witten equations and 4-manifold topology (pdf)

• Matilde Marcolli, Seiberg-Witten gauge theory, pdf

• piljin yi, Seiberg-Witten theory – with a view toward MQCD (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]

• Wikipedia, Seiberg-Witten invariants

A useful discussion of the physical origins of the Seiberg-Witten equations for mathematicians:

• Siye Wu, The Geometry and Physics of the Seiberg-Witten Equations, Progress in mathematics 205 (2002)

In relation to integrable systems:

• Alexander Gorsky, Igor Krichever, Andrei Marshakov, Andrei Mironov, Andrey Morozov, Integrability and Seiberg-Witten Exact Solution, Phys. Lett. B 355 (1995) 466-474 [arXiv:hep-th/9505035, doi:10.1016/0370-2693(95)00723-X]

Discussion of lifts of SW-invariants to M-theory:

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

Discussion as cohomological field theory:

• Shuhan Jiang, Jürgen Jost: Cohomological field theories and generalized Seiberg-Witten equations [arXiv:2407.04019]

• Shuhan Jiang: Cohomological Field Theories and Generalized Seiberg-Witten Equations, talk at CQTS (Feb 2025) [slides:pdf]

Lift to stable homotopy groups of spheres

A lift of Seiberg-Witten invariants to classes in circle group-equivariant stable homotopy groups of spheres (referred to in terms of stable Cohomotopy by some authors):

• Mikio Furuta: Stable-Homotopy version of Seiberg-Witten invariant, Max Planck Institute for Mathematics Preprint (1997) [pdf]

• Mikio Furuta: Monopole Equation and the 11/8-Conjecture, Mathematical Research Letters 8 (2001) 279-291 [doi:10.4310/MRL.2001.v8.n3.a5]

• Stefan Bauer, Mikio Furuta: A stable cohomotopy refinement of Seiberg-Witten invariants: I, Invent. math. 155 (2004) 1–19 [arXiv:math/0204340, doi:10.1007/s00222-003-0288-5]

• Stefan Bauer: A stable cohomotopy refinement of Seiberg-Witten invariants: II, Invent. math. 155 (2004) 21–40 [arXiv:math/0204267, doi:10.1007/s00222-003-0289-4]

Review:

• Ming Xu: The Bauer-Furuta invariant and a Cohomotopy refined Ruberman invariant, PhD thesis, Stony Brook (2004) [pdf, pdf]

• Tim Perutz: MO comment [MO:a/139921]

• Arun Debray: Furuta’s $10/8$ Theorem (2019) [pdf, pdf]

Further discussion:

• Mikio Furuta, Yukio Kametani, Hirofumi Matsue, Norihiko Minami: Homotopy theoretical considerations of the Bauer–Furuta stable homotopy Seiberg–Witten invariants, Geometry and Topology Monographs 10 (2007) 155-166 [doi:10.2140/gtm.2007.10.155, arXiv:0903.4462]

• Nobuhiro Nakamura: Bauer–Furuta invariants under $\mathbb{Z}_2$-actions, Math. Z. 262 (2009) 219–233 [doi:10.1007/s00209-008-0370-1]

• Chanyoung Sung: Equivariant Bauer-Furuta invariants on Some Connected Sums of 4-manifolds, Tokyo J. Math. 40 1 (2017) 53-63 [doi:10.3836/tjm/1502179215]

Seiberg-Witten curve

SW-curve in SW-theory

The notion of the Seiberg-Witten curve in Seiberg-Witten theory originates in:

• Nathan Seiberg, Edward Witten, pp. 37 in: Monopoles, duality and chiral symmetry breaking in $N=2$

  supersymmetric QCD, Nucl. Phys. B431 (1994) 484-550 [hep-th/9408099]

Review:

• Taro Kimura, §4.2.3.1 in: Instanton Counting, Quantum Geometry and Algebra, Mathematical Physics Studies, Springer (2021) [doi:10.1007/978-3-030-76190-5_4]

and in relation to integrable systems:

• Andrei Marshakov, Seiberg-Witten Curves and Integrable Systems [arXiv:hep-th/9903252]

As M5-brane worldvolume

Observation that under geometric engineering of D=4 N=2 SYM on D4/NS5 intersections and further lift to a single M5-brane, the SW-curve is identified with the M5-worldvolume transverse to the SYM-spacetime:

• 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]

Further early discussion:

• Karl Landsteiner, Esperanza Lopez, David A. Lowe, $\mathcal{N}$=2 Supersymmetric Gauge Theories, Branes and Orientifolds, Nucl. Phys. B 507 (1997) 197-226 [arXiv:hep-th/9705199, doi:10.1016/S0550-3213(97)00559-2]

• A. Brandhuber, Jacob Sonnenschein, Stefan Theisen, Shimon Yankielowicz, M Theory And Seiberg-Witten Curves: Orthogonal and Symplectic Groups, Nucl. Phys. B 504 (1997) 175-188 [arXiv:hep-th/9705232, doi:10.1016/S0550-3213(97)00531-2]

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]

• Francesco Benini, Sergio Benvenuti, Yuji Tachikawa, p. 2 of: Webs of five-branes and $\mathcal{N}=2$ superconformal field theories, JHEP 0909:052 (2009) [arXiv:0906.0359]

• 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]

Quantum SW-curve

On the “quantum SW curve”, a kind of quantized SW curve:

• Alba Grassi, Jie Gu, Marcos Mariño, Non-perturbative approaches to the quantum Seiberg-Witten curve, J. High Energ. Phys. 2020 106 (2020) [arXiv:1908.07065, doi:10.1007/JHEP07(2020)106 ]

In relation to class S-theories and “M3-brane”-defect branes inside M5-branes:

• Jin Chen, Babak Haghighat, Hee-Cheol Kim, Marcus Sperling, Elliptic Quantum Curves of Class $\mathcal{S}_k$, J. High Energ. Phys. 2021 28 (2021) [arXiv:2008.05155, doi:10.1007/JHEP03(2021)028]

In relation to E-strings and D6-D8-brane bound states:

• Jin Chen, Babak Haghighat, Hee-Cheol Kim, Marcus Sperling, Xin Wang, E-string Quantum Curve, Nuclear Physics B 973 (2021) 115602 [arXiv:2103.16996, doi:10.1016/j.nuclphysb.2021.115602]

On why the SW-curve should be quantized this way, as seen from topological string theory:

• Mina Aganagic, Robbert Dijkgraaf, Albrecht Klemm, Marcos Mariño, Cumrun Vafa, pp. 7 of: Topological Strings and Integrable Hierarchies, Commun. Math. Phys. 261 (2006) 451-516 [arXiv:hep-th/0312085, doi:10.1007/s00220-005-1448-9]

• Mina Aganagic, Miranda Cheng, Robbert Dijkgraaf, Daniel Krefl, Cumrun Vafa, §2 in: Quantum Geometry of Refined Topological Strings, J. High Energ. Phys. 2012 19 (2012) [arXiv:1105.0630, doi:10.1007/JHEP11(2012)019]

Relation to Rozansky-Witten invariants

On relation between Rozansky-Witten invariants and Seiberg-Witten invariants of 3-manifolds:

• Matthias Blau, George Thompson, On the Relationship between the Rozansky-Witten and the 3-Dimensional Seiberg-Witten Invariants, Adv. Theor. Math. Phys. 5 (2002) 483-498 [arXiv:hep-th/0006244]