Exact Lattice Supersymmetry at the Quantum Level for N=2 Wess-Zumino models in Lower Dimensions

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 operator 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.