Classification of affine vortices

Abstract

We prove a Hitchin–Kobayashi correspondence for vortices on the complex affine line with Kähler target, which generalizes a result of Taubes for the case of a line target. More precisely, suppose that $K$ is a compact Lie group and that the target $X$ is either a compact Kähler $K$-Hamiltonian manifold or $X$ is a symplectic vector space with linear $K$-action and a proper moment map. Suppose that the action of the complexified Lie group $G$ satisfies stable $=$ semistable. Then, for some sufficiently divisible integer $n$, there is a bijection between gauge equivalence classes of $K$-vortices with target $X$ and isomorphism classes of maps from the weighted projective line $\mathbb{P}(1,n)$ to $X/G$ that map the stacky point at infinity $\mathbb{P}\left(n\right)$ to the semistable locus of $X$. The results allow the construction and partial computation of the quantum Kirwan map from Woodward and play a role in the conjectures of Dimofte, Gukov, and Hollands relating vortex counts to knot invariants.