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Consider a Hamiltonian action of a compact connected Lie group on asymplectic manifold $(M,\omega)$. Conjecturally, under suitable assumptionsthere exists a morphism of cohomological field theories from the equivariantGromov-Witten theory of $(M,\omega)$ to the Gromov-Witten theory of thesymplectic quotient. The morphism should be a deformation of the Kirwan map.The idea, due to D. A. Salamon, is to define such a deformation by countinggauge equivalence classes of symplectic vortices over the complex plane $C$.  The present memoir is part of a project whose goal is to make this definitionrigorous. Its main results deal with the symplectically aspherical case. Thefirst one states that every sequence of equivalence classes of vortices overthe plane has a subsequence that converges to a new type of genus zero stablemap, provided that the energies of the vortices are uniformly bounded. Such astable map consists of equivalence classes of vortices over the plane andholomorphic spheres in the symplectic quotient. The second main result is thatthe vertical differential of the vortex equations over the plane (at the levelof gauge equivalence) is a Fredholm operator of a specified index.  Potentially the quantum Kirwan map can be used to compute the quantumcohomology of symplectic quotients.

A Quantum Kirwan Map: Bubbling and Fredholm Theory for Symplectic Vortices over the Plane