  Consider a Hamiltonian action of a compact connected Lie group $G$ on an
aspherical symplectic manifold $(M,\omega)$. Under some assumptions on
$(M,\omega)$ and the action, D. A. Salamon conjectured that counting gauge
equivalence classes of symplectic vortices on the plane $R^2$ gives rise to a
quantum deformation $Q\kappa_G$ of the Kirwan map. This article is the first of
three, whose goal is to define $Q\kappa_G$ rigorously. Its main result is that
the vertical differential of the vortex equations over $R^2$ (at the level of
gauge equivalence) is a Fredholm operator of a specified index. Potentially,
the map $Q\kappa_G$ 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 (see
[Woodward-Ziltener]).
