Abstract
The purpose of this paper is to describe a relationship between the moduli space of vortices and the moduli space of instantons. We study charge vortices in Yang-Mills-Higgs theories and show that the moduli space is isomorphic to a special lagrangian submanifold of the moduli space of instantons in non-commutative Yang-Mills theories. This submanifold is the fixed point set of a action on the instanton moduli space which rotates the instantons in a plane. To derive this relationship, we present a D-brane construction in which the dynamics of vortices is described by the Higgs branch of a gauge theory with 4 supercharges which is a truncation of the familiar ADHM gauge theory. We further describe a moduli space construction for semi-local vortices, lumps in the and grassmannian sigma-models, and vortices on the non-commutative plane. We argue that this relationship between vortices and instantons underlies many of the quantitative similarities between quantum field theories in two and four dimensions.