Intrinsic Geometry of Vortices

Abelian Higgs Vortices are solitons in two dimensions that exist on any Riemann surface. Particularly interesting mathematically is to consider vortices on a surface of constant curvature (of a particular magnitude). Here the vortex equations are formally integrable, and a vortex can be interpreted as defining a new metric with the same constant curvature but introducing additional conical singularities. The project aims to find intrinsic properties of these new metrics, for example the ratio of the geodesic separation of the vortex centres in the old and new metrics. It is hoped that Ed can better understand the vortex moduli space and its metric in terms of these intrinsic geometrical quantities. It is currently unknown if the vortex moduli spaces defined using field theory are isomorphic to the moduli spaces of surfaces with singularities
defined by complex geometers. Another question is whether all vortices correspond to branched coverings of the initial surface.

This project is in mathematical physics, with overlap to complex geometry, and is in the remit of EPSRC.