U(1)-vortices and quantum Kirwan map

TL;DR

This paper classifies finite energy solutions to the symplectic vortex equations over the complex plane with U(1) symmetry, extends known results, and computes the quantum Kirwan map for projective space.

Contribution

It generalizes Taubes' classification to higher dimensions and explicitly computes the quantum Kirwan map for complex projective spaces.

Findings

Classified all finite energy solutions to the vortex equations.

Described the moduli spaces and their compactifications.

Computed the quantum Kirwan map explicitly.

Abstract

We study the symplectic vortex equation over the complex plane, for the target space ${\mathbb C}^N$ ($N\geq 2$) with diagonal U(1)-action. We classify all solutions with finite energy and identify their moduli spaces, which generalizes Taubes' result for N=1. We also studied their compactifications and use them to compute the associated quantum Kirwan maps $\kappa_Q: H^*_{U(1)}({\mathbb C}^N) \to QH^*({\mathbb P}^{N-1})$.