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We construct a nonperturbative regularization for Euclidean noncommutativesupersymmetric Yang-Mills theories with four (N= (2,2)), eight (N= (4,4)) andsixteen (N= (8,8)) supercharges in two dimensions. The construction relies onorbifolds with discrete torsion, which allows noncommuting space dimensions tobe generated dynamically from zero dimensional matrix model in thedeconstruction limit. We also nonperturbatively prove that the twistedtopological sectors of ordinary supersymmetric Yang-Mills theory are equivalentto a noncommutative field theory on the topologically trivial sector withreduced rank and quantized noncommutativity parameter. The key point of theproof is to reinterpret 't Hooft's twisted boundary condition as an orbifoldwith discrete torsion by lifting the lattice theory to a zero dimensionalmatrix theory.Comment: 36 pages, references added, minor typos fixe

Regularization of non-commutative SYM by orbifolds with discrete torsion and SL(2,Bbb Z) duality