We investigate to derive off-shell invariant twisted super Yang-Mills for N=2 in 2-dimensions and N=4 in 4-dimensions with a central charge by super connection ansatz formalism. We find off-shell invariant N=2 algebra with and without an extra constr
aint in 2-dimensions. On the other hand in 4-dimensions we find off-shell invariant N=4 twisted SUSY algebra including one central charge always with a constraint.
We have recently proposed a new lattice SUSY formulation which has exact lattice supersymmetry for Wess-Zumino models in one and two dimensions for all N=2 supercharges. This formulation is non-local in the coordinate space but the difference operato
r satisfies the Leibniz rule on the newly defined star product. Here we show that this lattice supersymmetry is kept exact at the quantum level by investigating Ward-Takahashi identities up to two loop level.
We propose a new lattice superfield formalism in momentum representation which accommodates species doublers of the lattice fermions and their bosonic counterparts as super multiplets. We explicitly show that one dimensional $N=2$ model with interact
ions has exact supersymmetry on the lattice for all super charges with lattice momentum. In coordinate representation the finite difference operator is made to satisfy Leibnitz rule by introducing a non local product, the star product, and the exact lattice supersymmetry is realized. Supersymmetric Ward identities are shown to be satisfied at one loop level.
