Multivalued Wess-Zumino-Novikov functional and chiral anomaly in hydrodynamics

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 transport properties of Weyl fermions and, in the case of a charged fluid, exhibits the chiral anomaly. It is also consistent with Onsager's semiclassical quantization of circulation. Additionally, we discuss the hydrodynamic analog of instantons and related topological invariants.