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

TL;DR

This paper investigates how certain four-dimensional fermionic theories with global anomalies can be canceled by topological quantum field theories using symmetry extension, with applications to the Standard Model and dark matter sectors.

Contribution

It determines the minimal gauge groups for anomaly cancellation via symmetry extension in 4D fermionic theories, providing a new way to replace fermions with topologically ordered TQFTs.

Findings

Identified minimal gauge groups for anomaly cancellation in 4D fermionic theories.

Constructed anomaly-free TQFTs that replace Weyl fermions in the Standard Model.

Proposed topologically ordered dark sectors as alternatives to missing fermions.

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 $\nu_R$, 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.