A non-perturbative mixed anomaly and fractional hydrodynamic transport

Abstract

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 $${\Omega }_{5}^{{\text{Spin}}}$$(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 $${\Omega }_{d}^{{\text{Spin}}}$$(Bℤ2 × BU(1)) and $${\Omega }_{d}^{{{\text{Pin}}}^{-}}$$ (BU(1)) in all degrees d = 0 through 5.