  I construct a Lattice Gauge Theory (LGT) with discrete Z_2 structure group
and an equivariant BRST symmetry that is physically equivalent to the standard
SU(2)-LGT. The measure of this Z_2-LGT is invariant under all the discrete
symmetries of the lattice and its partition function does not vanish. The
Topological Lattice Theories (TLT) that localize on the moduli spaces are
explicitly constructed and their BRST symmetry is exhibited. The ghosts of the
Z_2-invariant local LGT are integrated in favor of a nonlocal bosonic measure.
In addition to the SU(2) link variables and the coupling g^2, this effective
bosonic measure also depends on an auxiliary gauge invariant site variable of
canonical dimension two and on a gauge parameter \alpha. The relation between
the expectation value of the auxiliary field, the gauge parameter \alpha and
the lattice spacing $a$ is obtained to lowest order in the loop expansion. In
four dimensions and the critical limit this expectation value is a physical
scale proportional to \Lambda_L in the gauge \alpha=g^2 (11-n_f)/24+ O(g^4).
Implications for the loop expansion of observables in such a critical gauge are
discussed.
