Classification of Affine Vortices

We prove a Hitchin-Kobayashi correspondence for affine vortices generalizing a result of Jaffe-Taubes [19] for the action of the circle on the complex line. Namely, suppose a compact connected Lie group K acts on a Kähler manifold X with proper moment map so that stable=semistable for the action of the complexified Lie group G and X is equivariantly convex at infinity. Then, for some sufficiently divisible integer n, there is a bijection between gauge equivalence classes of K-vortices with target X modulo gauge and isomorphism classes of maps from the weighted projective line P(1, n) to X/G that map the stacky point at infinity P(n) to semistable locus of X. The results allow the construction and partial computation of the quantum Kirwan map in Woodward [40], and play a role in the conjectures of Dimofte, Gukov, and Hollands [10] relating vortex counts to knot invariants.