Geometry and physics of ALX metrics in gauge theory

July 25 to July 29, 2022

at the

American Institute of Mathematics, San Jose, California

organized by

Laura Fredrickson, Akos Nagy, Steven Rayan, and Hartmut Weiss

Original Announcement

This workshop will be devoted to the geometric features and physical interpretations of certain 4-dimensional metrics arising from equations in gauge theory.

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:

Material from the workshop

A list of participants.

The workshop schedule.