Seiberg-Witten theory in nLab

Contents

Contents

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.

• 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=2N=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=2N=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, 𝒩=2\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 10/810/8 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 ℤ 2\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]

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]