# Workshop: Geometry of the Vortex Equations

**Date: **November 27 - 30, 2012

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45

**Organizer: **Nuno RomÃ£o

Two-dimensional gauge theories based on the vortex equations have experienced a revival in recent years. Their moduli spaces provide solid ground to examine several paradigmatic features of gauge theories: for instance, they support natural L^{^2}-geometries, illustrate correspondences of Hitchin-Kobayashi type, and provide examples of wall-crossing phenomena under certain deformations. Gauged vortices also play a role in effective models for a variety of physical phenomena, and they have been embedded in various ways into field theories in higher dimensions. Some recent developments have now related the vortex equations to a rather broad range of problems in geometry and topology, such as the construction of invariants of Hamiltonian actions, quantum cohomology, topological quantum field theories, BPS counting and knot homologies.

This workshop gathered researchers from different backgrounds working on various aspects of the vortex equations and their moduli spaces, with special emphasis on three main topics:

- geometry and topology of vortex moduli spaces;
- gauged Gromov-Witten theory;
- vortices and higher-dimensional field theories.

The meeting comprised three days dedicated to each of these themes, followed by a fourth day with a more interdisciplinary focus. The three thematic days started by an introductory extended lecture, according to the following plan:

- Steven Bradlow (Illinois): Vortices, principal pairs and moduli spaces in gauge theory;
- Christopher Woodward (Rutgers): Gauged Gromov-Witten invariants and applications;
- Sergei Gukov (Caltech): From vortex counting to knot homologies.

This event formed a focal point for the group "Geometry of Gauged Vortices" participating in the Junior Hausdorff Trimester Program "Mathematical Physics" at HIM.