A geometrical approach to N = 2 super Yang-Mills theory on the two dimensional lattice

We propose a discretization of two dimensional euclidean Yang-Mills theories with N = 2 supersymmetry which preserves exactly both gauge invariance and an element of supersymmetry. The approach starts from the twisted form of the continuum super Yang Mills action which we show may be written in terms of two real Kähler-Dirac fields whose components transform into each other under the twisted supersymmetry. Once the theory is written in this geometrical language it is straightforward to discretize by mapping the component tensor fields to appropriate geometrical structures in the lattice and by replacing the continuum exterior derivative and its adjoint by appropriate lattice covariant difference operators. The lattice action is local and possesses a unique vacuum state while the use of Kähler-Dirac fermions ensures the model does not exhibit spectrum doubling.