A Quantum Kirwan Map, II: Bubbling

Consider a Hamiltonian action of a compact connected Lie group $G$ on an aspherical symplectic manifold $(M,\omega )$. Under suitable assumptions, counting gauge equivalence classes of (symplectic) vortices on the plane $R^2$ conjecturally gives rise to a quantum deformation $Q{k}_G$ of the Kirwan map. This is the second of a series of articles, whose goal is to define $Q{k}_G$ rigorously. The main result is that every sequence of vortices with uniformly bounded energies has a subsequence that converges to a genus 0 stable map of vortices on $R^2$ and holomorphic spheres in the symplectic quotient. Potentially, the map $Q{k}_G$ can be used to compute the quantum cohomology of many symplectic quotients. Conjecturally it also gives rise to quantum generalizations of non-abelian localization and abelianization.