1. Oscar García-Prada: Geometry of vortices on Riemann surfaces
This series of lectures is devoted to the geometry of moduli spaces of vortices on compact Riemann surfaces. Vortices can be regarded as solutions to some gauge theoretic equations involving a unitary connection on a fibre bundle and a Higgs field, or alternatively as pairs consisting of a holomorphic bundle and a holomorphic section of some associated bundle. To establish the link, a stability condition is required on the Higgs pair. Themes to be treated include abelian and non-abelian vortices, dimensional reduction of instantons and vortices, vortices and Hodge bundles, the Kähler-Yang-Mills equations and gravitating vortices.
2. Niklas Garner: Categorical aspects of vortices
Vortex equations arise from supersymmetric quantum field theory (QFT) in describing certain BPS field configurations in three spacetime dimensions (or two spatial dimensions). In this minicourse, we will describe aspects of the corresponding QFTs and will extract from these vortex equations a category describing certain BPS line defects therein. Time permitting, we will use this physical setup to connect to notions in generalized affine Springer theory.
3. Nuno Romão: Quantization of vortices
Vortex moduli spaces support intrinsic geometry (their L^2 Kähler metric), which has been used to approximate the classical dynamics of vortices in low-energy field theory, e.g. via the associated geodesic flow. An extension of this idea is to use the same L^2 geometry to address the quantum mechanics of vortices in 2+1 dimensions via quantisation of their moduli spaces. There are various ways of doing this, depending on how the vortex dynamics is set up, and this series of lectures is meant to illustrate some of the possibilities. The holomorphic quantisation of the moduli space (relevant for first-order dynamics) will be discussed in the simplest example of vortices in line bundles. Some rudiments of the canonical quantisation of vortex moduli will also be presented, specifically the counting of states via L^2-Betti numbers, which is the viewpoint appropriate in a supersymmetric extension of second-order dynamics.
4. Martin Speight: L^2 geometry of moduli spaces of vortices and lumps
The low energy classical dynamics of topological solitons can often be modelled as geodesic motion in the space of static solitons with respect to a natural Riemannian metric called the L^2 metric. In this minicourse we will develop techniques to calculate this metric, and extract information about the resultant dynamics, for sigma model lumps and abelian vortices. A recurrent theme will be interesting phenomena arising due to noncompactness of the moduli space of static solitons.
5. Chris Woodward: Symplectic vortices and the quantum Kirwan map
This minicourse is devoted to the relationship between gauged pseudoholomorphic maps to a symplectic manifold with Hamiltonian group actions and the pseudoholomorphic maps to its symplectic quotient. We will discuss the large-area Gaio-Salamon adiabatic limit of gauged maps as well as the small-area limit in which the moduli space essentially reduces to a classical symplectic quotient. In the Gaio-Salamon limit the Ziltener compactification of the moduli space of affine vortices naturally arises and the quantum Kirwan map is defined by integration over the Ziltener compactification. We will explain the theory using the example of toric varieties as symplectic quotients of vector spaces.