and indices are well-known to be integral, since they merely count with signs the number of supersymmetric ground states. Note that the compactification breaks manifest modularity. Indeed, suppose we didn’t already know that $Z_{RR}({\mathcal F})$ was holomorphic (for ${\mathcal F}$ compact). Then this compactification implements the canonical way to extract a holomorphic function from a real-analytic modular form: analytically continue away from $\bar \tau = \tau^*$ and take the limit $\bar \tau \to -i \infty$.[^16]

Depending on the value of the gravitational anomaly $n$, one can make stronger statements than mere integrality. The $q$-dependence is immaterial — the statements hold in general for SQM models of degree $n$ equipped with a time-reversal symmetry.[^17] There are various ways to define the notion of “degree-$n$ SQM model”, just like the choices in §\[sec.anomaly\] for how to handle the gravitational anomaly. One general way is to work with SQM models that are relative, in the sense of [@FreedTeleman2012], to certain short-range-entangled $(1+1)$-dimensional phases. When $n\geq0$, another explicit method is to employ spectator fermions. Then a [*degree-$n$ SQM model*]{} is an SQM model (i.e. a super Hilbert space ${\mathcal H}$ with an odd operator $G$ generating the supersymmetry; it is “compact” when $G$ is 