ICTS

I will start by introducing the notion of Berry connection in quantum mechanics and quantum field theory — a connection on 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 K¨ahler 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.

We shall discuss construction and classification results for complete non-compact hyperk¨ahler 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 hyperk¨ahler 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.

I will start by introducing the notion of Berry connection in quantum mechanics and quantum field theory — a connection on 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 K¨ahler 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.

We shall discuss construction and classification results for complete non-compact hyperk¨ahler 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 hyperk¨ahler 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.

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.

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.

We shall discuss construction and classification results for complete non-compact hyperk¨ahler 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 hyperk¨ahler 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.

We shall discuss construction and classification results for complete non-compact hyperk¨ahler 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 hyperk¨ahler 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.

I will start by introducing the notion of Berry connection in quantum mechanics and quantum field theory — a connection on 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 K¨ahler 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.