Weak supersymmetric $su(N|1)$ quantum systems

We present several examples of supersymmetric quantum mechanical systems with weak superalgebra $su(N|1)$. One of them is the weak $su(N|1)$ oscillator. It has a singlet ground state, $N +1$ degenerate states at the first excited level, etc. Starting from the level $k = N+1$, the system has complete supersymmetric multiplets at each level involving $2^N$ degenerate states. Due to the fact that the supermultiplets are not complete for $k \leq N$, the Witten index represents a nontrivial function of $\beta$. 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 $N$ levels of the spectrum are not shifted, and we are dealing with a quasi-exactly solvable system. Another system represents a weak generalisation of the superconformal mechanics with $N$ complex supercharges. Also in this case, starting from a certain energy, the spectrum involves only complete supersymmetric $2^N$-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.)