1. Andrea Ferrari: Berry phases in supersymmetric QFT, periodic monopoles and generalised cohomology
I will start by introducing the notion of Berry connection in quantum mechanics and quantum field theory — a connection in a vector bundle of ground states fibering over deformation parameters. Then I shall explain that 2d supersymmetric theories quantised on a circle and endowed with a symmetry have Berry connections which are generalised periodic monopoles. As we will see, when these theories can be defined as sigma-models with Kähler manifolds as target, physical observables (such as the ground states) are related to (generalised) cohomology theories of these target manifolds. I will conclude this series of lectures by discussing two different kinds of spectral data of generalised periodic monopoles and, for Berry connections arising from sigma-models, by finally relating these data to the geometry of the target manifold.
2. Lorenzo Foscolo: Hypertoric geometry of Coulomb branches
We shall discuss construction and classification results for complete non-compact hyperkähler manifolds. We will focus on manifolds that admit an asymptotic geometry that generalises to higher dimensions the one of 4-dimensional ALF gravitational instantons such as the Taub-NUT and the Atiyah--Hitchin metrics. The 4-dimensional gravitational instantons have been recently classified and a relevant role in this theory is played by the Gibbons--Hawking ansatz, which allows one to describe 4-dimensional hyperkähler metrics with a circle symmetry explicitly. In higher dimensions, results are much more sporadic. In the minicourse we will discuss the role of higher dimensional generalisations of the Gibbons--Hawking ansatz, which can be used to construct examples with a torus symmetry, the so-called hypertoric varieties. We will also examine known examples arising from moduli spaces in gauge theory, such as monopoles, solutions to Nahm's equations and moduli spaces of instantons on ALF spaces. Finally, we will discuss how a large class of examples is expected to arise from Coulomb branches of 3-dimensional quantum gauge theory in theoretical physics.
3. Derek Harland: L2 geometry of hyperbolic monopole moduli
This minicourse introduces BPS monopoles on Euclidean space and hyperbolic space, and discusses their L2 geometry. Monopoles are examples of topological solitons with origins in particle physics. The L2 metrics on their moduli spaces can be used to study dynamics and are also geometrically interesting in their own right. This course will introduce some remarkable tools that can be used to construct monopoles, such as rational maps, spectral curves, and the Nahm transform. It will then review the L2 geometry of Euclidean monopoles, before looking at some recent work on the L2 geometry of hyperbolic monopoles.
4. Siqi He: The extended Bogomolny equation, Higgs bundles, and the Kobayashi--Hitchin correspondence
In this minicourse, we discuss the extended Bogomolny equations and their relation to Higgs bundles and the Kobayashi--Hitchin correspondence. These equations arise naturally as a dimensional reduction of the Kapustin--Witten equations and provide a bridge between gauge theory, complex geometry, and low-dimensional topology. We begin with a review of Higgs bundles on Riemann surfaces and the classical Kobayashi–Hitchin correspondence, which identifies stable Higgs bundles with solutions to Hitchin’s equations. We then introduce the extended Bogomolny equations on three-manifolds with boundary, emphasizing their geometric meaning and analytic structure. A central theme of the course is to explain how solutions to the extended Bogomolny equations can be interpreted in terms of holomorphic data, and how a version of the Kobayashi--Hitchin correspondence extends to this setting. We will also discuss boundary conditions, singular solutions, and their role in recent developments in gauge theory.
5. Chris Kottke: Analytical aspects of monopole moduli spaces
In this minicourse we will consider the asymptotic geometry of the L2 metrics on the moduli spaces of Euclidean BPS monopoles, emphasizing their "quasi-fibered-boundary" (QFB) structure. This QFB structure encodes a hierarchy of asymptotic regions of the moduli space, associated to cluster decompositions of monopoles with fixed charge into widely separated monopoles of lower charge. Accordingly, the metric admits asymptotics involving the metrics of lower charge moduli spaces. We will then discuss some implications for the analysis of harmonic forms on QFB manifolds with an eye toward Sen's conjecture for the L2 Hodge theory of monopole moduli spaces.