Weak supersymmetric su(N|1) quantum systems

Abstract

In this paper, we present several examples of supersymmetric quantum mechanical systems with weak superalgebra [Formula: see text]. One of them is the weak [Formula: see text] oscillator. It has a singlet ground state, [Formula: see text] degenerate states at the first excited level, etc. Starting from the level [Formula: see text], the system has complete supersymmetric multiplets at each level involving [Formula: see text] degenerate states. Due to the fact that the supermultiplets are not complete for [Formula: see text], the Witten index represents a nontrivial function of [Formula: see text]. This system can be deformed with keeping the algebra intact. The index is invariant under such deformation. The deformed system is not exactly solved, but the invariance of the index implies that the energies of the states at the first [Formula: see text] levels of the spectrum are not shifted, and we are dealing with a quasi-exactly solvable system. Another system represents a weak generalization of the superconformal mechanics with [Formula: see text] complex supercharges. Also in this case, starting from a certain energy, the spectrum involves only complete supersymmetric [Formula: see text]-plets. (There also exist normalizable states with lower energies, but they do not have normalizable superpartners. To keep supersymmetry, we have to eliminate these states.)