Abstract
We present a new non-perturbative 't Hooft anomaly afflicting a quantum field theory with symmetry group G=U(1)× Z₂ in four dimensions. We use the Adams spectral sequence to compute that the bordism group ΩSpin₅(BG), which classifies anomalies that remain when perturbative anomalies cancel, is Z₄. By constructing a mapping torus and evaluating the Atiyah-Patodi-Singer η-invariant, we show that the mod 4 anomaly is generated by a pair of Weyl fermions that are vector-like under U(1), but with only one component charged under Z₂. We construct a simple microscopic field theory that realises the anomaly, before investigating its impact in the hydrodynamic limit. We find that the anomaly dictates transport phenomena in the U(1) current and energy-momentum tensor akin to the chiral vortical and magnetic effects (even though the perturbative anomalies here vanish), but with the conductivities being fractionally quantised in units of a quarter, reflecting the mod 4 nature of the bordism group.Along the way, we compute the (relevant) bordism groups ΩSpin_d(BZ₂× BU(1)) and Ω_d^Pin⁻(BU(1)) in all degrees d=0 through 5.