U(1)-vortices and quantum Kirwan map

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 $\kappa_Q: H^*_{U(1)}({\mathbb C}^N) \to QH^*({\mathbb P}^{N-1})$.