Multivalued Wess-Zumino-Novikov Functional and Chiral Anomaly in Hydrodynamics

TL;DR

This work addresses how to embed the chiral anomaly of Weyl fermions into hydrodynamics without introducing new spacetime scales. It achieves this by extending the perfect-fluid action with a multivalued Wess-Zumino-Novikov functional, introducing a chiral phase $\Theta$ and a helicity current, and deriving a gauge-invariant particle current $I^\mu=n^\mu+\tfrac{k}{2}h^\mu$ whose divergence encodes the anomaly. The framework yields a modified continuity equation $\partial_\mu n^\mu=-\tfrac{k}{4}\Omega_{\mu\nu}{}^*\Omega^{\mu\nu}$ (or $-\tfrac{k}{4}F_{\mu\nu}^*F^{\mu\nu}$ with external fields), together with an unchanged stress tensor, and a topological quantization condition $k\in\mathbb{Z}$ linked to the Hopf-Novikov structure and vortex-instanton dynamics. The approach connects hydrodynamics to Weyl-fermion physics, reproducing chiral transport phenomena such as the chiral vortical and magnetic effects and providing a spin-channel interpretation via $\sigma^\mu=\tfrac12\Sigma^\mu$, with spin-vorticity coupling $k\Sigma^\mu\Omega_{\mu\nu}$ in Newtonian form, thus offering a topologically grounded, metric-free description of anomaly-enabled fluid dynamics.

Abstract

We present a hydrodynamic framework derived from the action of a perfect fluid, modified by the hydrodynamic analog of Novikov's multivalued functional. This modification introduces spin degrees of freedom into the fluid. The structure closely resembles the Abelian version of the Wess-Zumino functional, commonly applied in field theories with chiral anomalies. The deformation incorporates the transport properties of Weyl fermions and exhibits the chiral anomaly in the case of a charged fluid. It is also consistent with Onsager's semiclassical quantization of circulation. Additionally, we discuss the hydrodynamic analog of instantons and related topological invariants.