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

We construct super Yang-Mills theories with ${\cal N}=2, 4$ supersymmetrieson the two-dimensional square lattice keeping one or two supercharges exactly.Along the same line as the previous paper \cite{sugino}, the construction isbased on topological field theory formulation. In order to resolve the problemof degenerate classical vacua encountered in the previous paper, we considertwo kinds of modifications of the action. For one of them, any supersymmetrybreaking terms do not need to be introduced, and the formulations exactlyrealize some of supersymmetries at the lattice level. Our lattice actions flowto the desired continuum theories without any fine-tuning of parameters.Comment: 15 pages, no figure, LaTeX, (v2) published version in JHEP, (v3) minor corrections, (v4) 22 pages, reconsidered the vacuum degeneracy problem for SU(N), revised versio