  We construct a mathematical framework for twisted N=2 supersymmetric
topological quantum field theory on a 4-manifold. Supersymmetry in flat space
is defined and the twist homomorphism is constructed, giving us a supermanifold
that is the total space of an odd vector bundle over the even 4-manifold. A
special category of connections on this space is defined and a decomposition
into so-called component fields is proved. The twisted supersymmetric action is
computed, and the structure of the action, the decomposition, and the action of
a special odd vector field are all shown to have a rich geometrical structure
that was partially interpred by Atiyah and Jeffrey. In short, the action is an
infinite-dimensional analogue of the Euler class of the vector bundle of
self-dual 2-forms over the space of connections mod gauge. This geometrical
insight serves two purposes: first, it motivates the study of anti-self-dual
connections, intersection theory, and the action of the group of gauge
transformations, all of which appear by themselves after the twist. Secondly,
it sets the stage for an eventual proof of Witten's Conjecture, relating the
Donaldson and Seiberg-Witten invariants. What we build here amounts to a
mathematical treatment of a physical treatment of a mathematical construction
of Donaldson.
