Geometry and physics of ALX metrics in gauge theory

Original Announcement

These 4-dimensional 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 4-manifolds, 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 4-manifolds 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 quasi-ALX or QALX space (where X = E, F, of G) is a higher-dimensional generalization of an ALX space where certain singular behaviors are allowed.

These spaces have central roles in low-dimensional 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 Kapustin--Witten 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 4-manifold is closed and $\theta \neq 0, \tfrac{\pi}{2}$, then the Kapustin--Witten equations do not have interesting, smooth solutions. Thus one needs to consider noncompact manifolds or singular solutions. Natural choices for noncompact 4-manifolds 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 quasi-ALG. 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 one-size-fits-all 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.

• Kapustin--Witten equations on closed Kähler surfaces and ALX manifolds. As mentioned in the introduction, very little is known about Kapustin--Witten 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 Kapustin--Witten 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.

• Hitchin--Kobayashi correspondence for the Kapustin--Witten equations. Over compact Kähler surfaces, the structure of the $\theta = 0$ Kapustin--Witten 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 Kapustin--Witten moduli embed into the moduli space of Higgs bundles (holomorphic bundles with endomorphism-valued holomorphic 1-forms) over the Kähler surface. We propose investigating the properties of this embedding, mainly whether it has an inverse under certain circumstances. In some instances, Hitchin--Kobayashi-style correspondences have been obtained for reduced Kapustin-Witten equations on noncompact 3-manifolds; 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.

Material from the workshop

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Workshop Videos

Papers arising from the workshop: