Abstract
Starting from the geometrical interpretation of integrable vortices on two-dimensional hyperbolic space as conical singularities, we explain how this picture can be expressed in the language of Cartan connections, and how it can be lifted to the double cover of three-dimensional Anti-de Sitter space viewed as a trivial circle bundle over hyperbolic space. We show that vortex configurations on the double cover of AdS space give rise to solutions of the Dirac equation minimally coupled to the magnetic field of the vortex. After stereographic projection to (2+1)-dimensional Minkowski space we obtain, from each lifted hyperbolic vortex, a Dirac field and an abelian gauge field which solve a Lorentzian, (2+1)-dimensional version of the Seiberg–Witten equations.
Highlights
Vortex configurations consisting of a complex scalar and an abelian gauge field can be given a geometrical interpretation in two dimensions by viewing the modulus of the scalar field as a conformal rescaling of the underlying two-dimensional geometry
We showed that the equivariant formulation of the vortex equation on S3 can be solved in terms of bundle maps of the Hopf fibration, and that each vortex configuration gives rise to a solution of the gauged Dirac equation on S3 and, after stereographic projection, on Euclidean space
We show that the hyperbolic vortex equations can be interpreted as the flatness conditions for a su(1, 1) Cartan connection which encodes the geometry, modified by the vortices
Summary
Vortex configurations consisting of a complex scalar and an abelian gauge field can be given a geometrical interpretation in two dimensions by viewing the modulus of the scalar field as a conformal rescaling of the underlying two-dimensional geometry. We showed that the equivariant formulation of the vortex equation on S3 can be solved in terms of bundle maps of the Hopf fibration, and that each vortex configuration gives rise to a solution of the gauged Dirac equation on S3 and, after stereographic projection, on Euclidean space. In this way, equivariant Popov vortices on SU (2) can be seen to generate manifestly square integrable and smooth solutions of the magnetic zero-mode problem posed and studied by Loss and Yau [5].
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