Chiral Metric Hydrodynamics, Kelvin Circulation Theorem, and the Fractional Quantum Hall Effect

By extending the Poisson algebra of ideal hydrodynamics to include a two-index tensor field, we construct a new (2+1)-dimensional hydrodynamic theory that we call "chiral metric hydrodynamics." The theory breaks spatial parity and contains a degree of freedom which can be interpreted as a dynamical metric, and describes a medium which behaves like a solid at high frequency and a fluid with odd viscosity at low frequency. We derive a version of the Kelvin circulation theorem for the new hydrodynamics, in which the vorticity is replaced by a linear combination of the vorticity and the dynamical Gaussian curvature density. We argue that the chiral metric hydrodynamics, coupled to a dynamical gauge field, correctly describes the long-wavelength dynamics of quantum Hall Jain states with filling factors $\nu=N/(2N+1)$ and $\nu=(N+1)/(2N+1)$ at large $N$. The Kelvin circulation theorem implies a relationship between the electron density and the dynamical Gaussian curvature density. We present an purely algebraic derivation of the low-momentum asymptotics of the static structure factor of the Jain states.