  It is known that certain theories with extended supersymmetry can be
discretized in such a way as to preserve an exact fermionic symmetry. In the
simplest model of this kind, we show that this residual supersymmetric
invariance is actually a BRST symmetry associated with gauge fixing an
underlying local shift symmetry. Furthermore, the starting lattice action is
then seen to be entirely a gauge fixing term. The corresponding continuum
theory is known to be a topological field theory. We look, in detail, at one
example - supersymmetric quantum mechanics which possesses two such BRST
symmetries. In this case, we show that the lattice theory can be obtained by
blocking out of the continuum in a carefully chosen background metric. Such a
procedure will not change the Ward identities corresponding to the BRST
symmetries since they correspond to topological observables. Thus, at the
quantum level, the continuum BRST symmetry is preserved in the lattice theory.
Similar conclusions are reached for the two-dimensional complex Wess-Zumino
model and imply that all the supersymmetric Ward identities are satisfied {\it
exactly} on the lattice. Numerical results supporting these conclusions are
presented.

High Energy Physics - Lattice;High Energy Physics - Theory