$\operatorname{Cliff}(n,{{\mathds}C})$-module, and so the index is automatically even. In summary, the index of a degree-$n$ SQM model with time reversal symmetry lives in $m{{\mathds}Z}$ with: $$\label{eqn.mofn}
 m = \begin{cases} 
 1, & n = 0 \pmod 8, \\
 2, & n = 4 \pmod 8, \\
 0, & \text{else}.
\end{cases}$$ In the SQFT case that we care about, the Witten genus $\eta^n Z_{RR}({\mathcal F})$ has $q$-expansion in $m{{\mathds}Z}(\!(q)\!)$.[^20]

### The mod-2 index

That the indexes of time-reversal SQM models vanish in degrees $2$ and $6$ mod $8$ and are even in degree $4$ mod $8$ is compensated by a more refined “mod-2 index,” which is nontrivial in degrees $n=1$ and $2 \pmod 8$. We will review its construction because it already measures some torsion in the space of SQFTs and because a variation of the mod-2 index appears when trying to understand indexes of noncompact SQM models. We start with the case $n=1$. The even subalgebra of $\operatorname{Cliff}(1,{{\mathds}R})$ is isomorphic to ${{\mathds}R}$, and $\operatorname{Cliff}(1,{{\mathds}R})$ has a unique irreducible module, namely itself. We will call it ${{\mathds}R}^{1|1}$. The supersymmetric ground states $V$ of a degree-$1$ SQM model is then isomorphic to ${{\mathds}R}^{a|a}$ for some nonnegative integer $a$. By definition, the [*mod-2 index*]{} 