one-point function of } \bar{G}  \mbox{ in ${\mathcal B}$}).$$

A simple generalization of this construction is to require ${\mathcal B}_p$ to stabilize to some compact SQFT ${\mathcal B}_+$ for large positive $p$ and to another compact SQFT ${\mathcal B}_-$ for large positive $p$ for large negative $p$. Then we expect, up to a phase factor: $$\begin{gathered}
\label{eqn.bordismZRR}
\sqrt{-8\tau_2} \frac{\partial}{\partial\bar{\tau}}\bigl[\text{Witten genus of }{\mathcal F}\bigr]  = \bigl[ (\text{torus one-point function of } {:}\psi_1\cdots\psi_n{:}\bar{G}  \mbox{ in ${\mathcal B}_+$}) \\  -(\text{torus one-point function of } {:}\psi_1\cdots\psi_n{:}\bar{G}  \mbox{ in ${\mathcal B}_-$}) \bigr]\end{gathered}$$ This means that, although the torus one-point function of $\bar{G}$ is not a deformation-invariant of an SQFT, it changes in a controlled fashion. This control is the basis of our torsion invariant.

Integrality of the $q$-expansion {#sec.int}
--------------------------------

In addition to holomorphicity, the other fundamental fact about the Witten genus $\eta^n Z_{RR}({\mathcal F})$ of a (compact) SQFT ${\mathcal F}$ is the integrality of its $q$-expansion. We briefly review the argument. Let ${\mathcal F}[S^1]$ denote the $S^1$-equivariant supersymmetric quantum mechanics model produced by compactifying ${\mathcal F}$ on a circle (with Ramond aka nonbounding spin structure). Then $\eta^n Z_{RR}({\mathcal F})$ can be interpreted as the supersymmetric index of ${\mathcal F}[S^1]$, with $q$ parameterizing the $S^1$-action, 