Various Super Yang-Mills Theories with Exact Supersymmetry on the Lattice

We continue to construct lattice super Yang-Mills theories along the linediscussed in the previous papers \cite{sugino, sugino2}. In our construction of${\cal N}=2, 4$ theories in four dimensions, the problem of degenerate vacuaseen in \cite{sugino} is resolved by extending some fields and soaking upwould-be zero-modes in the continuum limit, while in the weak couplingexpansion some surplus modes appear both in bosonic and fermionic sectorsreflecting the exact supersymmetry. A slight modification to the models is madesuch that all the surplus modes are eliminated in two- and three-dimensionalmodels obtained by dimensional reduction thereof. ${\cal N}=4, 8$ models inthree dimensions need fine-tuning of three and one parameters respectively toobtain the desired continuum theories, while two-dimensional models with ${\calN}=4, 8$ do not require any fine-tuning.Comment: 28 pages, no figure, LaTeX, JHEP style; (v2) published version to JHEP; (v3) argument on the vacuum degeneracy revised, 34 page