Classification of affine vortices

TL;DR

This paper establishes a correspondence between affine vortices and certain maps to quotient stacks, extending known results and enabling new computations in quantum geometry and knot theory.

Contribution

It generalizes the Hitchin-Kobayashi correspondence for affine vortices to broader settings involving Hamiltonian actions and weighted projective lines.

Findings

Proves a bijection between gauge classes of vortices and maps to quotient stacks.

Enables construction and partial computation of the quantum Kirwan map.

Connects vortex counts to knot invariants in conjectural frameworks.

Abstract

We prove a Hitchin-Kobayashi correspondence for affine vortices generalizing a result of Jaffe-Taubes for the action of the circle on the affine line. Namely, suppose a compact Lie group K has a Hamiltonian action on a Kaehler manifold X which is either compact or convex at infinity with a proper moment map, and so that stable=semistable for the action of the complexified Lie group G. 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 the semistable locus in X. The results allow the construction and partial computation of the quantum Kirwan map in Woodward, and play a role in the conjectures of Dimofte, Gukov, and Hollande relating vortex counts to knot invariants.