Twisting to Abelian BF/Chern-Simons Theories

Starting from a $D=3$, $N=4$ supersymmetric theory for matter fields, a twist with a Grassmann parity change is defined which maps the theory into a gauge fixed, abelian $BF$ theory on curved 3-manifolds. After adding surface terms to this theory, the twist is seen to map the resulting supersymmetric action to two uncoupled copies of the gauge fixed Chern-Simons action. In addition, we give a map which takes the $BF$ and Chern-Simons theories into Donaldson-Witten TQFT's. A similar construction, but with $N=2$ supersymmetry, is given in two dimensions.