deformation invariant. The same is true when $n = 5$, $6$, or $7 \pmod 8$.

### Noncompact SQM models

We turn now to the index of “mildly noncompact” SQM models. The definition of “mild noncompactness” mirrors §\[sec.noncompact\]: the SQM model ${\mathcal X}$ should come with a local operator $\Phi$ parameterizing the noncompact direction; writing ${\mathcal Y}_p$ for the theory produced from ${\mathcal X}$ by adding a fermion $\lambda$ and a fermionic superpotential $\lambda(\Phi-p)$, we demand that ${\mathcal Y}_p$ stabilizes in the limits $p \to \pm \infty$ to compact SQM models ${\mathcal Y}_\pm$. For definiteness, we will first describe the case when ${\mathcal X}$ has degree $n=4$, in which case each ${\mathcal Y}_p$ is an SQM model of degree $3$.

We lose no generality by assuming that as $p$ varies, the Hilbert space ${\mathcal H}$ of ${\mathcal Y}_p$ is independent of $p$, with the only variation being in the choice of supersymmetry.[^22] Since ${\mathcal Y}_p$ has degree $3$, ${\mathcal H}$ is naturally a module for $\operatorname{Cliff}(3,{{\mathds}R})$. Choose[^23] an isomorphism $\operatorname{Cliff}(3,{{\mathds}R}) \cong {{\mathds}H}\otimes \operatorname{Cliff}(-1,{{\mathds}R})$, and let $\gamma$ denote the generator of $\operatorname{Cliff}(-1,{{\mathds}R})$. The supersymmetry generator in ${\mathcal Y}_p$ is $G(p) = g(p) \gamma$, where $g(p)$ is a quaternionic matrix; the time-reversal structure 