Supersymmetric Yang-Mills quantum mechanics in various dimensions

Supersymmetric Yang-Mills quantum mechanics (SYMQM) results from the dimensional reduction of the Yang-Mills field theory in $D$ space-time dimensions to a single point in the $D-1$ dimensional space. It can be also viewed as the effective quantum mechanics of zero momentum modes of the original theory. These systems were first considered in 80's as simple models with supersymmetry. Independently, zero-volume field theories (especially pure Yang-Mills) were employed as the starting point of the small volume expansion, which is an important theoretical tool complementary to early lattice calculations. Later the models attracted a new wave of interest after the hypothesis of the equivalence, between the $D=10, SU(\infty)$ SYMQM and a M-theory of D0 branes. In this talk some new results obtained for the D=4 system with SU(2) gauge symmetry will be presented.