Vafa–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 Vafa–Witten twist is then the splitting ${\displaystyle (\mathbf {1} ,\mathbf {2} )\oplus (\mathbf {1} ,\mathbf {2} )}$.^[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 Vafa–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 Kapustin–Witten twist (${\displaystyle (\mathbf {2} ,\mathbf {1} )\oplus (\mathbf {1} ,\mathbf {2} )}$).^[1]

The bosonic fields of Vafa–Witten theory are a gauge field ${\displaystyle A}$ with field strength ${\displaystyle F}$, a scalar field ${\displaystyle \Phi }$ and a self-dual 2-form ${\displaystyle B}$. The Vafa–Witten equations are then given by:^[1]

${\displaystyle F_{\mu \nu }^{+}+{\frac {1}{2}}[C,B_{\mu \nu }^{+}]+{\frac {1}{4}}[B_{\mu \kappa }^{+},B_{\nu \lambda }^{+}]g^{\kappa \lambda }=0;}$

${\displaystyle D_{\mu }C+D^{\nu }B_{\mu \nu }^{+}=0.}$

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

• Tanaka, Yuuji (2013). "Some boundedness properties of solutions to the Vafa-Witten equations on closed four-manifolds". Q. J. Math. 68: 1203–1225. arXiv:1308.0862. doi:10.1093/qmath/hax015.
• Tanaka, Yuuji (2014). "A perturbation and generic smoothness of the Vafa-Witten moduli spaces on closed symplectic four-manifolds". Glasg. Math. J. 61: 471–486. arXiv:1410.1691. doi:10.1017/S0017089518000307.
• Clifford Henry Taubes (2017). "The behavior of sequences of solutions to the Vafa-Witten equations". arXiv:1702.04610.{{cite arXiv}}: CS1 maint: missing class (link) A bot will complete this citation soon. Click here to jump the queue
• Manshot, Jan (2017). "Vafa-Witten theory and iterated integrals of modular forms". Communications in Mathematical Physics. 371: 787–831. arXiv:1709.10098. doi:10.1007/s00220-019-03389-5.
• Manshot, Jan (2023). "Four-Manifold Invariants and Donaldson-Witten Theory". arXiv:2312.14709.{{cite arXiv}}: CS1 maint: missing class (link) A bot will complete this citation soon. Click here to jump the queue