Variational principle for free surface boundary dynamics of dissipationless fluids with Hall viscosity

In this talk, we present the variational principle for non-dissipative and compressible two-dimensional broken parity fluids with odd viscosity with a free surface boundary condition. In the absence of shear viscosity, the boundary layer is formed by odd viscosity coupling the divergence of velocity to vorticity.
The dissipationless nature of the boundary layer allows us to construct an effective action that gives both bulk hydrodynamic equations and free surface boundary conditions. The free surface boundary conditions require an additional boundary term in the action which resembles a 1+1D chiral boson field coupled to the background geometry. We discuss generalizations of this action to quantum Hall fluids and nearly incompressible fluids.