From Twisted Supersymmetry to Orbifold Lattices

We show how to derive the supersymmetric orbifold lattices of Cohen et al. \cite{Cohen:2003xe,Cohen:2003qw} and Kaplan et al. \cite{Kaplan:2005ta} by direct discretization of an appropriate twisted supersymmetric Yang-Mills theory. We examine in detail the four supercharge two dimensional theory and the theory with sixteen supercharges in four dimensions. The continuum limit of the latter theory is the well known Marcus twist of ${\cal N}=4$ Yang-Mills. The lattice models are gauge invariant and possess one exact supersymmetry at non-zero lattice spacing.