Anomaly of 4d Weyl fermion with discrete symmetries

TL;DR

This work develops a unified, computable framework for 4d Weyl fermion anomalies with discrete charges by descending from continuous $Spin\times U(1)$ and $Spin^c$ theories via cobordism. It computes explicit anomaly indices for ${\rm Spin}\times\mathbb{Z}_n$ and ${\rm Spin}\times_{\mathbb{Z}_2^{\mathrm{F}}}\mathbb{Z}_{2m}$ through $\eta$-invariants evaluated on carefully chosen 5d manifold generators, and expresses the results using the ${\rm TP}_5$ cobordism groups and their images. The paper furnishes concrete formulas for the discrete anomalies, introduces new discrete manifold generators, and demonstrates how these indices control nonperturbative global anomalies in theories with discrete gauge/global symmetries, with applications to discrete extensions of the Standard Model and topological dark matter. Overall, it provides a rigorous topological route to anomaly cancellation and classification in discrete symmetry settings, with broad implications for model building and beyond-Standard-Model physics. The results offer a principled way to analyze and cancel discrete anomalies in fermionic theories and connect to recent advances in topological phases and cobordism-based classifications.

Abstract

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.