Vortices, Instantons and Branes

TL;DR

This work uncovers a deep correspondence between the moduli spaces of vortices in U(N) Yang–Mills–Higgs theories and instantons in non-commutative Yang–Mills, showing that the vortex moduli space ${\cal V}_{k,N}$ is a complex middle-dimensional (special Lagrangian) submanifold of the instanton moduli space ${\cal I}_{k,N_f}$, realized as the fixed point set of a holomorphic U(1) action. A D-brane construction identifies ${\cal V}_{k,N}$ with the Higgs branch of a U(k) theory, while the ADHM framework for instantons provides a precise embedding via a symmetry deformation. The paper extends the map to semi-local vortices and non-commutative settings, revealing topology-changing transitions and a Seiberg-like duality in the vortex sector. These results illuminate why 2D and 4D SUSY theories display shared non-perturbative structures, and they offer a unified geometric lens for studying BPS states across dimensions.

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.

Figures (4)

• Figure 1: The brane configuration for $U(N)$ gauge theory with $N$ hypermultiplets. Figure 1A shows the theory on the Coulomb branch. In Figure 1B, the theory has a FI parameter and lies in its unique ground state. The D1-branes are the vortices.
• Figure 2: The Eguchi-Hanson Christmas cracker. The D4-branes are wrapped around the shaded region.
• Figure 3: The brane configuration for $U(N)$ gauge theory with $N+M$ hypermultiplets, and $k$ vortices.
• Figure 4: The brane configuration for vortices on a non-commutative background. A NS-NS B field lies in the $x^1-x^2$ directions, inducing non-commutivity on the D3-brane worldvolume and causing the D1-brane to tilt.