Hydrodynamics of two-dimensional CFTs

We demonstrate that the geometric action on a coadjoint orbit of the Virasoro group appropriately describes non-dissipative two-dimensional conformal fluids. While this action had already appeared in the context of AdS$_3$ gravity, the hydrodynamical interpretation given here is new. We use this to argue that the geometric action manifestly controls both sides of the fluid/gravity correspondence, where the gravitational `hologram' gives an effective hydrodynamical description of the dual CFT. As a byproduct, our work sheds light on the nature of the AdS$_3$ reparametrization theory used to effectively compute Virasoro vacuum blocks at large central charge, as the reparametrization mode is now understood as a fluctuation of the fluid velocity.