  We construct a nonperturbative regularization for Euclidean noncommutative
supersymmetric Yang-Mills theories with four (N= (2,2)), eight (N= (4,4)) and
sixteen (N= (8,8)) supercharges in two dimensions. The construction relies on
orbifolds with discrete torsion, which allows noncommuting space dimensions to
be generated dynamically from zero dimensional matrix model in the
deconstruction limit. We also nonperturbatively prove that the twisted
topological sectors of ordinary supersymmetric Yang-Mills theory are equivalent
to a noncommutative field theory on the topologically trivial sector with
reduced rank and quantized noncommutativity parameter. The key point of the
proof is to reinterpret 't Hooft's twisted boundary condition as an orbifold
with discrete torsion by lifting the lattice theory to a zero dimensional
matrix theory.
