A disorder-averaged framework for evaluating the Witten index

We study supersymmetric indices in disordered systems, in particular an $\mathcal{N}=4$ supersymmetric Sachdev-Ye-Kitaev-type quantum mechanics. In cases where the disordered parameters do not affect the index, we explain how the exact answer can nevertheless be obtained using disorder averaged collective variables. Furthermore, when disorder averaging is performed on multiple copies of the same theory, non-trivial coupling between them is generated. We show how the index ultimately remains factorized.