Relationship between various supersymmetric lattice models

We comment on the relationships between several supersymmetric lattice models; the 'orbifold lattice theory' by Cohen-Kaplan-Katz-Unsal (CKKU), lattice regularization of the topological field theory by Sugino and the 'geometrical approach' by Catterall. We point out that these three models have close relationships; the N = (2, 2) model by Catterall and the two-dimensional N = (2, 2) lattice theory being similar to Sugino's constructioncan be derived by appropriate truncation of fields in the two-dimensional N = (4, 4) orbifold lattice theory by CKKU [3]. Catterall's N = (2, 2) description possesses extra degrees of freedom compared to the target N = (2, 2) theory. If we remove those extra degrees of freedom in a way keeping supersymmetry on the lattice, Catterall's description reduces to a model of the Sugino type