Anomaly of 4d Weyl fermion with discrete symmetries

TL;DR

This paper derives explicit formulas for anomalies of 4d Weyl fermions under discrete symmetries by reducing known anomaly indices and evaluating eta-invariants on specific manifolds, providing a unified framework for classifying global anomalies.

Contribution

It introduces a systematic reduction method for known anomaly indices to discrete symmetries and explicitly computes the images of topological phase homomorphisms using eta-invariants on key manifolds.

Findings

Explicit anomaly index formulas for 4d Weyl fermions with discrete symmetries.

A unified framework for computing nonperturbative global anomalies.

Evaluation of eta-invariants on specific manifolds for anomaly classification.

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 $\eta$-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.