Anomaly of 4d Weyl fermion with discrete symmetries

We derive explicit anomaly index formulas for 4d Weyl fermions charged under discrete symmetries $\mathrm{Spin} \times \mathbb{Z}_n$ and $\mathrm{Spin} \times_{\mathbb{Z}_2^{\mathrm{F}}} \mathbb{Z}_{2m}$ by systematically reducing the known perturbative local anomaly indices for $\mathrm{Spin} \times \mathrm{U}(1)$ and $\mathrm{Spin} \times_{\mathbb{Z}_2^{\mathrm{F}}} \mathrm{U}(1) \equiv \mathrm{Spin}^c$ symmetries. Our approach leverages the natural group homomorphisms $\mathrm{TP}_5(\mathrm{Spin} \times \mathrm{U}(1)) \longrightarrow \mathrm{TP}_5(\mathrm{Spin} \times \mathbb{Z}_n), \quad \mathrm{TP}_5(\mathrm{Spin}^c) \longrightarrow \mathrm{TP}_5(\mathrm{Spin} \times_{\mathbb{Z}_2^{\mathrm{F}}} \mathbb{Z}_{2m}),$ which map continuous topological phases to their discrete counterparts. We compute the images of these homomorphisms explicitly by evaluating $η$-invariants on key manifold generators: the 5d lens space bundle $X(n;1,1)$ and $L(n;1) \times \mathrm{K3}$ (with $\mathrm{K3}$ the K3 surface) for $\mathrm{Spin} \times \mathbb{Z}_n$, and the 5d lens space $L(m;1,1,1)$ and $L(m;1) \times \mathrm{E}$ (with $\mathrm{E}$ the Enriques surface) if $m$ is even, or $L(m;1) \times \mathrm{K3}$ if $m$ is odd for $\mathrm{Spin} \times_{\mathbb{Z}_2^{\mathrm{F}}} \mathbb{Z}_{2m}$. These results provide a unifying framework to compute and classify nonperturbative global anomalies of discrete symmetries -- in fermionic theories with discrete gauge or global symmetries.

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