Anomalous (3+1)d Fermionic Topological Quantum Field Theories via Symmetry Extension

TL;DR

The paper develops a framework to cancel nonperturbative global anomalies of 3+1d Weyl fermions carrying discrete symmetries by extending the global symmetry G to a larger group G_tot through a finite normal subgroup K. It leverages Atiyah-Patodi-Singer eta invariants to compute anomaly indices in the 5D cobordism group TP5 and applies the symmetry-extension criterion r^*ν_G = 0 to realize a (3+1)d anomalous K-gauge TQFT that matches the original anomaly on the boundary. It provides explicit anomaly indices for Spin-based and Spin×Z_n/G-extensions, discusses applications to the Standard Model with missing ν_R, and presents concrete examples and general theorems showing when and how symmetry extension can trivialize fermionic anomalies for various 2-adic, 3-adic, and coprime torsion cases. The results suggest a mechanism by which a gapped, topologically ordered dark sector or an anomalous TQFT can replace missing Weyl fermions without invoking conventional Higgs mechanisms, with quantified minimal gauge groups K for different symmetry settings. Overall, the work offers a principled approach to reconcile fermionic global anomalies with symmetric, nontrivial topological orders in (3+1)d field theories and beyond."

Abstract

Discrete finite-group global symmetries may suffer from nonperturbative 't-Hooft anomalies. Such global anomalies can be canceled by anomalous symmetry-preserving topological quantum field theories (TQFTs), which contain no local point operators but only extended excitations such as line and surface operators. In this work, we study mixed gauge-gravitational nonperturbative global anomalies of Weyl fermions (or Weyl semimetals in condensed matter) charged under discrete Abelian internal symmetries in four-dimensional spacetime, with spacetime-internal fermionic symmetry $G=$Spin$\times_{\mathbb{Z}_2^{\rm F}}\mathbb{Z}_{2m}^{\rm F}$ or Spin$\times\mathbb{Z}_n$ that contains fermion parity $\mathbb{Z}_{2}^{\rm F}$. We determine the minimal finite gauge group $K$ of anomalous $G$-symmetric TQFTs that can match the fermionic anomaly via the symmetry-extension construction $1 \to K \to G_{\rm Tot} \to G \to 1$, where the anomaly in $G$ is trivialized upon pullback to $G_{\rm Tot}$, computed by Atiyah-Patodi-Singer eta invariant. This allows one to replace a $G$-symmetric four-dimensional Weyl fermion by an anomalous $G$-symmetric discrete-$K$-gauge TQFT as an alternative low-energy theory in the same deformation class. As an application, we show that the four-dimensional Standard Model with 15 Weyl fermions per family, in the absence of a sterile right-handed neutrino $ν_R$, exhibits mixed gauge-gravitational global anomalies between baryon and lepton number symmetries $({\bf B \pm L})$ and spacetime diffeomorphisms. We identify the corresponding minimal $K$-gauge fermionic TQFT that cancels these anomalies and can be interpreted as a gapped, topologically ordered dark sector replacing missing Weyl fermions via symmetry extension, without invoking conventional Anderson-Higgs symmetry breaking.