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

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${\times }_{{\mathbb{Z}}_2^{\mathrm{F}}}{\mathbb{Z}}_{2m}^{\mathrm{F}}$ or Spin$\times {\mathbb{Z}}_n$ that contains fermion parity ${\mathbb{Z}}_2^{\mathrm{F}}$. We determine the minimal finite gauge group $K$ of anomalous $G$-symmetric TQFTs that can match the fermionic anomaly via the symmetry-extension construction $1\to K\to G_{\mathrm{Tot}}\to G\to 1$, where the anomaly in $G$ is trivialized upon pullback to $G_{\mathrm{Tot}}$, 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 $(\mathbf{B}\pm \mathbf{L})$ 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.

Comments: 28 pages