Normalized vacuum states in N=4 supersymmetric Yang-Mills quantum mechanics with any gauge group

We study the question of existence and the number of normalized vacuum stat es in N = 4 super-Yang-Mills quantum mechanics for any gauge group. The mas s deformation method is the simplest and dearest one. It allowed us to calc ulate the number of normalized vacuum states for all gauge groups, For all unitary groups, #(vac) = 1, but for the symplectic groups [starting from Sp (6) ], for the orthogonal groups [starting from SO(8)] and for all the exce ptional groups, it is greater than one, We also discuss at length the funct ional integral method. We calculate the "deficit term" for some non-unitary groups and predict the value of the integral giving the "principal contrib ution". The issues like the Born-Oppenheimer procedure to derive the effect ive theory and the manifestation of the localized vacua in the asymptotic e ffective wave functions are also discussed. (C) 2000 Elsevier Science B.V. All rights reserved.