When a $4D$ supersymmetric theory is placed on $S^3 \times \mathbb{R}$, the supersymmetric algebra is necessarily modified to $su(2|1)$ (its central extension) and we are dealing with a {\it weak} supersymmetric system. For such systems, the excited states of the Hamiltonian are not all paired. As a result, the Witten index Tr$\{(-1)^F e^{-\beta H}\}$ is no longer an integer number, but a $\beta$-dependent function. This index is often called ``superconformal index", because it is useful for studying dualities in superconformal gauge theories, but conformal symmetry plays no role in its definition.
Similarly to the ordinary Witten index, this index stays invariant under deformations of the theory that keep the supersymmetry algebra intact. Based on the Hilbert space analysis, we give a simple general proof of this fact. We then show how this invariance works for two simplest weak supersymmetric quantum mechanical systems involving a real or a complex bosonic degree of freedom.
There exist also weak supersymmetric quantum mechanical systems based on the central extension of the algebra $su(N|1)$ with arbitrary $N$.
We present a couple of examples of such systems and discuss their dynamics.