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 is a compact Lie group and that the target is either a compact Kähler -Hamiltonian manifold or is a symplectic vector space with linear -action and a proper moment map. Suppose that the action of the complexified Lie group satisfies stable semistable. Then, for some sufficiently divisible integer , there is a bijection between gauge equivalence classes of -vortices with target and isomorphism classes of maps from the weighted projective line to that map the stacky point at infinity to the semistable locus of . 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.
"Classification of affine vortices." Duke Math. J. 165 (9) 1695 - 1751, 15 June 2016. https://doi.org/10.1215/00127094-3450315