Kapustin–Witten equations

A topological twist is a choice for how the four-dimensional spin group ${\displaystyle \operatorname {Spin} (4)=\operatorname {SU} (2)\times \operatorname {SU} (2)}$ acts on the canonical inclusion representation ${\displaystyle \mathbf {4} \colon \operatorname {SU} (4)\hookrightarrow \operatorname {GL} _{4}(\mathbb {C} )}$ of the R-symmetry group, which relates the topology to the supersymmetry. The Kapustin-Witten twist is then the splitting ${\displaystyle (\mathbf {2} ,\mathbf {1} )\oplus (\mathbf {1} ,\mathbf {2} )}$.^[1]

In 1988, Edward Witten developed topological quantum field theory (TQFT),^[2] a contribution listed for his Fields Medal in 1990,^[3] and used it to showed that Donaldson theory, a contribution listed for Simon Donaldson's Fields Medal in 1986,^[4] is a TT N = 2 SYM. In 1994, Nathan Seiberg and Edward Witten then constructed the dual TT N = 2 SYM, known as Seiberg–Witten theory.^[5][6] Both theories were very successful in describing smooth 4-manifolds. Later development then shifted from the N = 2 case with two inequivalent twists to the N = 4 case with three inequivalent twists. Besides the Kapustin–Witten twist, these are the Donaldson–Witten twist (${\displaystyle (\mathbf {1} ,\mathbf {2} )\oplus (\mathbf {1} ,\mathbf {1} )\oplus (\mathbf {1} ,\mathbf {1} )}$) and the Vafa–Witten twist (${\displaystyle (\mathbf {1} ,\mathbf {2} )\oplus (\mathbf {1} ,\mathbf {2} )}$).^[1]

The bosonic fields of Kapustin–Witten theory are a gauge field ${\displaystyle A}$ with field strength ${\displaystyle F}$ and 1-form ${\displaystyle \phi }$. The Kapustin-Witten equations are then given by:^[1]

${\displaystyle (F-\phi \wedge \phi +tD\phi )^{+}=0;}$

${\displaystyle (F-\phi \wedge \phi -t^{-1}D\phi )^{-}=0;}$

${\displaystyle D^{*}\phi =0.}$

• N = 1 supersymmetric Yang–Mills theory
• Seiberg–Witten equations

• Kapustin, Anton; Witten, Edward (2007). "Electric-magnetic duality and the geometric Langlands program". Communications in Number Theory and Physics. 1 (1): 1–236. arXiv:hep-th/0604151. Bibcode:2007CNTP....1....1K. doi:10.4310/cntp.2007.v1.n1.a1. S2CID 30505126.
• Kapustin, Anton (2007). "Langlands Duality and Topological Field Theory" (PDF). icmat.es. Retrieved 2026-08-01.
• Witten, Edward (2008). "Mirror Symmetry, Hitchin's Equations, and Langlands Duality". The Many Facets of Geometry. pp. 113–128. arXiv:0802.0999. doi:10.1093/acprof:oso/9780199534920.003.0007. ISBN 978-0-19-953492-0.
• Manshot, Jan (2023). "Four-Manifold Invariants and Donaldson-Witten Theory". arXiv:2312.14709 [hep-th].