Dirac-K\"{a}hler fermions and exact lattice supersymmetry

We discuss a new approach to putting supersymmetric theories on the lattice. The basic idea is to start from a {\it twisted} formulation of the underlying supersymmetric theory in which the fermions are represented as grassmann valued antisymmetric tensor fields. The original supersymmetry algebra is replaced by a twisted algebra which contains a scalar nilpotent supercharge $Q$. Furthermore the action of the theory can then be written as the $Q$-variation of some function. The case of ${\cal N}=2$ super Yang-Mills theory in two dimensions is discussed in some detail. We then present our proposal for discretizing this theory and derive the resultant lattice action. The latter is local, free of spectrum doubling, gauge invariant and preserves the scalar supercharge invariance exactly. Some preliminary numerical results are then presented. The approach can be naturally generalized to yield a lattice action for ${\cal N}=4$ super Yang-Mills in four dimensions.