Topology of vortices with toric targets

Abstract

The vortex equations are PDEs describing BPS field configurations for gauged sigma models in fibre bundles. In the work surveyed here, we focus on the situation where the base is a Riemann surface and the fibre/target is a toric Kähler manifold, both assumed compact. Then the moduli spaces of solutions (up to unitary gauge) are generalised configuration spaces associated to a certain simplicial complex that can be extracted from the toric data. The fundamental groups of such spaces can be understood
as a novel type of surface braid groups with coloured strands, which we shall describe in very concrete terms.