Consider a Hamiltonian action of a compact connected Lie group on a symplectic manifold \((M,\omega)\). Conjecturally, under suitable assumptions there exists a morphism of cohomological field theories from the equivariant Gromov-Witten theory of \((M,\omega)\) to the Gromov-Witten theory of the symplectic 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 counting gauge equivalence classes of symplectic vortices over the complex plane \(\mathbb{C}\).

The present memoir is part of a project whose goal is to make this definition rigorous. Its main results deal with the symplectically aspherical case.