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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.

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      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


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          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


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              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.