
## 1. Kapustin–Witten equations

Kapustin and Witten [MR2306566] constructed a family of "twisted" $\mathcal{N} = 4$ gauge theories in four dimensions in order to build a bridge between gauge theory and the geometric Langlands correspondence. The solutions to the Kapustin–Witten family of equations on closed 4-manifolds are not very interesting, since away from the special values $t=0,\infty$, these are complex flat connections not depending on $t$, while when $t=0,\infty$ and $M$ is Kähler, they reduce to Simpson’s equations by a result of Yuuji Tanaka [MR3928797].

For the more interesting case of either non-compact manifolds, or manifolds with boundary, it is thus desirable to understand and formulate meaningful conjectures about the geometry of their spaces of solutions. In the first case, a result of Akos Nagy and Goncalo Oliveira [MR4280510] shows that no solutions with finite energy can exist on ALE and ALF spaces modeled on $\mathbb{R}^{4}$ or $\mathbb{R}^{3}\times S^{1}$, since the finite energy condition forces the Higgs field to vanish, leading to instantons.
1. ### New directions in Kapustin–Witten theory

The Kapustin–Witten equations on closed Kähler and ALX 4-dimensional manifolds are a source of interesting geometry whose state of the art is rapidly evolving. It is thus desirable to discuss the relation between the Kapustin–Witten equations with complex anti-self-dual field configurations, Higgs bundles, and the Vafa–Witten equations.

#### Problem 1.1.

[Steven Rayan] $(i)$ To uncover new properties of the Kapustin–Witten equations. For instance, is there a Hitchin–Kobayashi interpretation, or stability type results?

$(ii)$ Use the Hitchin equations as a paradigm to uncover the Kapustin–Witten equations in 4 real dimensions. In particular, in the case when the underlying 4-manifold is an ALX space, what properties are recovered by the resulting moduli spaces?
• ### Geometry of moduli spaces of solutions

#### Problem 1.2.

[Panagiotis Dimakis, Akos Nagy, Steve Rayan] $(i)$ To discuss ways to produce solutions with possibly infinite energy on ALX spaces. Can one find solutions to the dimensionally reduced equations and lift them in a nontrivial way?

$(ii)$ When $t=1$, the Kapustin–Witten and Vafa–Witten equations are reductions of the same 5-dimensional Haydys–Witten equations. How can the moduli spaces of solutions be related?

$(iii)$ For manifolds with boundary, determine how the prescription of (generalized) Nahm pole asymptotics along the boundary, relates solutions to the Kapustin–Witten equations to Witten’s approach to the Jones polynomial.

$(iii)$ Determine the curvature properties of the $L^{2}$-metric, as well as it asymptotic properties at infinity.
• ### Geometric realization of heterotic duality

#### Problem 1.3.

[Gonçalo Oliveira, Song Sun] To determine a geometric realization of heterotic duality. Do the connection and $B$-field in the Kapustin–Witten equations correspond to a structure coming from a monopole?

Cite this as: AimPL: Geometry and physics of ALX metrics in gauge theory, available at http://aimpl.org/geomphysalx.