A non-perturbative mixed anomaly and fractional hydrodynamic transport

A bstract We present a new non-perturbative ’t Hooft anomaly afflicting a quantum field theory with symmetry group G = U (1) × ℤ 2 in four dimensions. We use the Adams spectral sequence to compute that the bordism group ₅^{Spin} (BG), which classifies anomalies that remain when perturbative anomalies cancel, is ℤ 4. 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 ℤ 2. 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 ₃^{Spin} (B ℤ 2 × B U (1) ) and ₃^{{Pin}^-} (B U (1) ) in all degrees d = 0 through 5.