Extended Supersymmetry and Super-BF Gauge Theories

The absence of fermion kinetic terms in supersymmetric-BF gauge theories is established. We do this by means of explicit off-shell (superspace) constructions. As part of our study we give the superspace constraints for D=3, N=4 super Yang-Mills along with the D=3, N=4 superconformal algebra. The puzzle we are interested in solving is the fact that the topological cousins, known as super-BF gauge theories, of certain supersymmetric-BF theories have kinetic terms for the twisted fermions. We show that the map which takes the latter to the former includes a Hodge decomposition of the twisted fermions. In conjunction with this result, we argue that it is natural to modify the naive path integral measure of supersymmetric-BF theories to include the Ray-Singer analytic torsion.