Super Yang-Mills Theories on the Two-Dimensional Lattice with Exact Supersymmetry

We construct super Yang-Mills theories with ${\cal N}=2, 4$ supersymmetries on the two-dimensional square lattice keeping one or two supercharges exactly. Along the same line as the previous paper \cite{sugino}, the construction is based on topological field theory formulation. In order to resolve the problem of degenerate classical vacua encountered in the previous paper, we consider two kinds of modifications of the action. For one of them, any supersymmetry breaking terms do not need to be introduced, and the formulations exactly realize some of supersymmetries at the lattice level. Our lattice actions flow to the desired continuum theories without any fine-tuning of parameters.