Anomaly of 4d Weyl fermions with discrete symmetries

We derive explicit anomaly-index formulas for four-dimensional Weyl fermions charged under the finite symmetries $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}\times {\mathbb{Z}}_n$ and $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}{\times }_{{\mathbb{Z}}_2^{\mathrm{F}}}{\mathbb{Z}}_{2m}^{\mathrm{F}}$. The strategy is to start from the standard perturbative anomaly indices for $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}\times \mathrm{U}(1)$ and $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}{\times }_{{\mathbb{Z}}_2^{\mathrm{F}}}\mathrm{U}(1)={\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^c$, and then restrict the continuous $\mathrm{U}(1)$ symmetry to a finite cyclic subgroup. On the level of invertible field theories this gives natural homomorphisms $${\mathrm{T}\mathrm{P}}_5(\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}\times \mathrm{U}(1))⟶{\mathrm{T}\mathrm{P}}_5(\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}\times {\mathbb{Z}}_n),{\mathrm{T}\mathrm{P}}_5({\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^c)⟶{\mathrm{T}\mathrm{P}}_5(\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}{\times }_{{\mathbb{Z}}_2^{\mathrm{F}}}{\mathbb{Z}}_{2m}^{\mathrm{F}})\mathrm{.}$$ We compute these maps explicitly by evaluating reduced $\eta$-invariants on geometric representatives of the finite anomaly groups. For $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}\times {\mathbb{Z}}_n$, the relevant backgrounds are the five-dimensional lens-space bundle $X(n;1,1)$ and the product $L(n;1)\times \mathrm{K}3$. For $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}{\times }_{{\mathbb{Z}}_2^{\mathrm{F}}}{\mathbb{Z}}_{2m}^{\mathrm{F}}$, the relevant backgrounds are $L(m;1,1,1)$ and, depending on the parity of $m$, either $L(m;1)\times \mathrm{E}\mathrm{n}\mathrm{r}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{s}$ or $L(m;1)\times \mathrm{K}3$. The output is a pair of integer-valued anomaly indices for each finite symmetry. These indices are normalized in the cyclic factors of the finite anomaly group, so they can be used directly in anomaly-cancellation checks for fermions with discrete gauge or global symmetries.