nLab Vafa-Witten equations

Idea

The Vafa-Witten twist is one of three inequivalent topological twists in topologically twisted D=4 super Yang-Mills theory, the other two being the Donaldson-Witten twist and the Kapustin-Witten twist (or geometric Langlands twist). The Vafa-Witten equations are the equations obtained from the Vafa-Witten twist.

Formulation

Consider

• $G$ a Lie group and $\mathfrak{g}$ its Lie algebra,

• $X$ an orientable Riemannian 4-manifold with Riemannian metric $g$ and volume form $vol_g$,

• $P\twoheadrightarrow X$ a principal $G$-bundle and $Ad(P)\coloneqq P\times_G\mathfrak{g}\twoheadrightarrow X$ its adjoint bundle,

• $A\in\Omega^1(X,Ad(P))$ a principal connection,

• $F_A\coloneqq\mathrm{d}A+\frac{1}{2}[A\wedge A]\in\Omega^2(X,Ad(P))$ its curvature and $F_A^+=\frac{1}{2}(F_A+\star F_A)$ its self-dual part,

• $a\in\Omega_+^2(X,Ad(P))$ a self-dual form.

The Vafa-Witten equations are:

$$\mathrm{d}_A a =0;$$

$$F_A^+ =\frac{1}{8}[a\bullet a].$$

(Clifford 17, Eq. (1), Guan 22C, Eq (1.3))

For $a=0$, the Vafa-Witten equations reduce to the self-dual Yang-Mills equations $F_A^+=0$.

It is also possible to consider a generalization with a smooth section $\Phi\in\Gamma^\infty(X,Ad(P))$ given by:

$$\mathrm{d}_A a +\star\mathrm{d}_A\Phi =0;$$

$$F_A^+ =\frac{1}{8}[a\bullet a] +\frac{1}{2}[\Phi,a].$$

(Tanaka 13, p. 2, Tanaka 14, Eq. (1.1) & (1.2), Clifford 17, Eq. (1.6), Manshot 23, Eq. (3.3), Guan 22A, Eq (1.1), Guan 22B, Eq (1.1) , Guan 22C, Eq (1.1), Dai & Guan 2025, Eq (1.1))

References

• Yuuji Tanaka, Some boundedness properties of solutions to the Vafa-Witten equations on closed four-manifolds (2013), arXiv:1308.0862
• Yuuji Tanaka, A perturbation and generic smoothness of the Vafa–Witten moduli spaces on closed symplectic four-manifolds (2014), arXiv:1410.1691
• Clifford Taubes, The behavior of sequences of solutions to the Vafa-Witten equations (2017), arXiv:1702.04610
• Ren Guan, On the general part of the perturbed Vafa-Witten moduli spaces on 4-manifolds (2022), arXiv:2207.03701
• Ren Guan, A variation of the reduced Vafa-Witten equations on 4-manifolds (2022), arXiv:2208.06131
• Ren Guan, Transversality of the perturbed reduced Vafa-Witten moduli spaces on 4-manifolds (2022), arXiv:2212.01586
• Jan Manshot, Four-Manifold Invariants and Donaldson-Witten Theory (2023), arXiv:2312.14709
• Bo Dai and Ren Guan, A simple perturbation of Vafa-Witten equations and a transversality result (2025), arXiv:2505.14702