Vortices, Instantons and Branes

TL;DR

This paper explores the deep mathematical relationship between vortex moduli spaces and instanton moduli spaces in gauge theories, revealing a geometric isomorphism and its implications for quantum field theories.

Contribution

It establishes an isomorphism between vortex and instanton moduli spaces using D-brane constructions and explores related moduli space structures for various vortex types.

Findings

Vortex moduli space is a special Lagrangian submanifold of instanton moduli space.

The relationship is derived via a D-brane construction involving a U(k) gauge theory.

This connection explains similarities between 2D and 4D quantum field theories.

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 k vortices in U(N) Yang-Mills-Higgs theories and show that the moduli space is isomorphic to a special Lagrangian submanifold of the moduli space of k instantons in non-commutative U(N) Yang-Mills theories. This submanifold is the fixed point set of a U(1) 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 U(k) 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 CP(N) 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 shared by quantum field theories in two and four dimensions.