Let $(M, \omega,G,\mu)$ be a symplectic manifold with a Hamiltonian action. Let $X$ be its symplectic reduction. The symplectic vortices on $M$ were introduced by Salamon, Mundet i Riera and etc 20 years ago. It is used to construct the so-called Hamiltonian Gromov-Witten invariants. Essentially, this is a new type of the Gromov-Witten theory for the reduction $X$ using the equivariant topological data of $M$. In this talk, I will review the topic following this line with $L^2$-moduli spaces of symplectic vortices. Furthermore, we generalize the vortex equation and introduce a new equivariant moduli space to give an equivariant Gromov-Witten theory for $M$ when $G$ is abelian. Combine these constructions, we may realize a quantum version of Kirwan map. The talk is mainly based on the joint work with Bai-Ling Wang and Rui Wang.