for SQM models of degree $3$ requires that $g(p)$ be “quaternionically self-adjoint,” meaning that its eigenvalues live in ${{\mathds}R}\subset {{\mathds}H}$. Thus, after a $p$-dependent change of basis, the only thing varying with $p$ is the spectrum of $g(p)$, which by compactness is a discrete subset (with finite multiplicities) of ${{\mathds}R}$.

The “index” of a noncompact SQM model ${\mathcal X}$ can be defined as the Ramond partition function $Z_R({\mathcal X}) = \mathrm{Tr}_{\mathcal H}(-1)^F \cdots$, but the noncompactness means that this partition function will depend nontrivially on the length of the worldline torus. The limit $\bar\tau \to -i\infty$ corresponds to the IR limit of $Z_R({\mathcal X})$, which merely counts supersymmetric ground states. We will use the term “index” to mean this IR limit.

If the limits $g(\pm\infty) = \lim_{p \to \pm \infty} g(p)$ have no kernel, then the index (in the IR sense) of ${\mathcal X}$ is relatively easy to compute: it counts with signs the number of eigenvalues of $g$ that cross $0$, times a factor of 2 coming from the quaternionic nature of degree-$3$ and degree-$4$ SQM models. Indeed, to say that the limits $g(\pm\infty)$ have no kernel is to say that supersymmetry is spontaneously broken in these limits, and 