Axial-Current Anomaly in Euler Fluids.

We argue that a close analog of the axial-current anomaly of quantum field theories with fermions occurs in the classical Euler fluid. The conservation of the axial current (closely related to the helicity of inviscid barotropic flow) is anomalously broken by the external electromagnetic field as ∂_{μ}j_{A}^{μ}=2E·B, similar to that of the axial current of a quantum field theory with Dirac fermions, such as QED.

Quantum hydrodynamics, the quantum benjamin-ono equation, and the Calogero model.

Collective field theory for the Calogero model represents particles with fractional statistics in terms of hydrodynamic modes--density and velocity fields. We show that the quantum hydrodynamics of this model can be written as a single evolution equation on a real holomorphic Bose field--the quantum integrable Benjamin-Ono equation. It renders tools of integrable systems to studies of nonlinear dynamics of 1D quantum liquids.

Chiral Nonlinear sigma models as models for topological superconductivity.

We study the mechanism of topological superconductivity in a hierarchical chain of chiral nonlinear sigma models (models of current algebra) in one, two, and three spatial dimensions. The models illustrate how the 1D Fröhlich's ideal conductivity extends to a genuine superconductivity in dimensions higher than one. The mechanism is based on the fact that a pointlike topological soliton carries an electric charge.

Integrable structure of interface dynamics.

We establish the equivalence of 2D contour dynamics to the dispersionless limit of the integrable Toda hierarchy constrained by a string equation. Remarkably, the same hierarchy underlies 2D quantum gravity.