A Non-Perturbative Mixed Anomaly and Fractional Hydrodynamic Transport

Abstract

We present a new non-perturbative 't Hooft anomaly afflicting a quantum field theory with symmetry group $G=U(1)\times {\mathbb{Z}}_2$ 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 ${\mathbb{Z}}_4$. By constructing a mapping torus and evaluating the Atiyah-Patodi-Singer $\eta$-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 ${\mathbb{Z}}_2$. 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 ${\Omega }_d^{\text{Spin}}(B{\mathbb{Z}}_2\times BU(1))$ and ${\Omega }_d^{{\text{Pin}}^{-}}(BU(1))$ in all degrees $d=0$ through 5.

Comments: 21+9 pages , 9 figures, comments are welcome