Witten index for weak supersymmetric systems: Invariance under deformations

Abstract

When a 4D supersymmetric theory is placed on [Formula: see text], the supersymmetric algebra is necessarily modified to [Formula: see text] and we are dealing with a weak supersymmetric system. For such systems, the excited states of the Hamiltonian are not all paired. As a result, the Witten index [Formula: see text] is no longer an integer number, but a [Formula: see text]-dependent function. However, this function 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.