Lattice supersymmetry in 1D with two supercharges

A consistent formulation of a fully supersymmetric theory on the lattice has been a long standing challenge. In recent years there has been a renewed interest on this problem with different approaches. At the basis of the formulation we present in the following there is the Dirac-Kahler twisting procedure, which was proposed in the continuum for a number of theories, including N=4 SUSY in four dimensions. Following the formalism developed in recent papers, an exact supersymmetric theory with two supercharges on a one dimensional lattice is realized using a matrix-based model. The matrix structure is obtained from the shift and clock matrices used in two dimensional non-commutative field theories. The matrix structure reproduces on a one dimensional lattice the expected modified Leibniz rule. Recent claims of inconsistency of the formalism are discussed and shown not to be relevant.