for this workshop
Geometry and physics of ALX metrics in gauge theory
at the
American Institute of Mathematics, San Jose, California
organized by
Laura Fredrickson, Akos Nagy, Steven Rayan, and Hartmut Weiss
This workshop, sponsored by AIM and the NSF, will be devoted to the geometric features and physical interpretations of certain 4dimensional metrics arising from equations in gauge theory.
These 4dimensional metrics are classically known within the differential geometry community as ALE, ALF, ALG, and ALH gravitational instantons and their generalizations. There are four important families of noncompact, complete Riemannian 4manifolds, called Asymptotically Locally Euclidean or ALE, Asymptotically Locally Flat or ALF, ALG, and ALH (the last two named unimaginatively via induction). While the precise definition of these spaces vary by authors, they are always noncompact 4manifolds with fibered ends. More precisely, the dimensions of the fiber are 0 (ALE), 1 (ALF), 2 (ALG), and 3 (ALH). The size of the fibers is bounded, and the volume growth of the manifold is $r^{4\dim (\mathrm{fiber})}$. An ALX space (where X = E, F, G, or H) is called a gravitational instanton if it is hyperkähler and satisfies certain curvature decay requirements. A quasiALX or QALX space (where X = E, F, of G) is a higherdimensional generalization of an ALX space where certain singular behaviors are allowed.
These spaces have central roles in lowdimensional gauge theories. Over the past thirty years, moduli spaces of vector bundles, monopoles, vortices, flat connections, and Higgs bundles have been incredibly influential in geometry. In particular, Higgs bundles have been studied in differential geometry, mathematical physics, and even in number theory through its connections to the Langlands program. The $L^2$ geometry of the Hitchin moduli spaces corresponding to closed Riemann surfaces is still a topic of intense research. For example, the asymptotics of the Hitchin metric have just recently been rigorously established. The end of the Hitchin moduli is a singular torus fibration, and, in fact, it is conjectured to be a QALG space with respect to this fibration.
It is still unclear what the appropriate generalization of Hitchin's equations is in higher dimensions. One candidate in 4D is the KapustinWitten equations. These are a family of equations parametrized by a real number $\theta \in [0, \pi)$. It is easy to see that if the underlying 4manifold is closed and $\theta \neq 0, \tfrac{\pi}{2}$, then the KapustinWitten equations do not have interesting, smooth solutions. Thus one needs to consider noncompact manifolds or singular solutions. Natural choices for noncompact 4manifolds are the ALE, ALF, ALG, and ALH spaces. Taubes has considered the ALE case recently. Furthermore, there have been projects studying noncompact manifolds with cylindrical ends and with singular boundary conditions. Beyond these results, little is known even about the analytic properties of these solutions and the geometry of moduli spaces over different manifolds.
The main topics are:

Verify that the $L^2$metric on the Higgs bundle
moduli space is QALG. It has long been a folklore expectation that
the natural hyperkähler metric on the moduli space of Higgs
bundles is quasiALG. The global properties of the metric are at
the heart of a number of conjectures related, for instance, to the
convergence of certain path integrals from physics defined over the
moduli space. Recent results establish some asymptotics of the
metric, at least in low rank away from certain ''bad'' loci.
However, there is still no onesizefitsall analytic approach to
understanding the asymptotics of the Hitchin metric. In particular,
we wish to understand the nature of the metric nearby to the
discriminant locus of the moduli space.
 KapustinWitten equations on closed Kähler surfaces
and ALX manifolds. As mentioned in the introduction, very little
is known about KapustinWitten fields on closed manifolds, and on
ALX spaces. We propose here the start of the study of such
problems. More concretely, we propose looking at the cases when the
underlying manifold, $X$, is either a closed Kähler surface, or an
ALX manifold (a ''gravitational instanton''). As a first step, the
group could establish basic existence and vanishing results. For
example, using dimensional reduction, one can construct solution
with spherical symmetry, when $X$ is a product of a closed surface
and $\mathbb{P}^1$ (in which case the KapustinWitten equation
reduces to Manton's five vortex equations). Furthermore, Taubes'
result gives some insight to flat factors of $\mathbb{R}^4$, which
are examples of (flat) ALX manifolds. Thus, building on Taubes
theorem, one might be able to prove nonexistence of nontrivial
solutions in those cases.
 HitchinKobayashi correspondence for the KapustinWitten equations. Over compact Kähler surfaces, the structure of the $\theta = 0$ KapustinWitten equations can be interpreted similarly to Hitchin's equations: a pair of equations consisting of a moment map equation on the curvature and the harmonicity of the Higgs field. It follows that KapustinWitten moduli embed into the moduli space of Higgs bundles (holomorphic bundles with endomorphismvalued holomorphic 1forms) over the Kähler surface. We propose investigating the properties of this embedding, mainly whether it has an inverse under certain circumstances. In some instances, HitchinKobayashistyle correspondences have been obtained for reduced KapustinWitten equations on noncompact 3manifolds; the extent to which this is true in general (and for the nonreduced case) is a key problem in complex gauge theory. \item Other problems to discuss may include noncollapsed limits of Einstein $4$manifolds, which are Einstein orbifolds with ALE bubbles. The question of which of these orbifolds can be desingularized or obtained as a limit is open. On the other hand, there are open questions in ALF geometry related to collapsed limits.
This event will be run as an AIMstyle workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
For more information email workshops@aimath.org