Fermionic dark matter

_Tex: 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×ℤF2ℤF2m or Spin×ℤn that contains fermion parity ℤF2. 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→K→GTot→G→1, where the anomaly in G is trivialized upon pullback to GTot, 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 νR, exhibits mixed gauge-gravitational global anomalies between baryon and lepton number symmetries (B±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.