In this talk I consider Berry phases that appear when a primary state in a 2D CFT undergoes adiabatic conformal transformations. Despite the infinite-dimensional parameter space, these phases can be evaluated exactly and coincide with the symplectic fluxes on Virasoro orbits computed long ago by Alekseev and Shatashvili. In 3D gravity, such phases provide a gravitational extension of Thomas precession. They also appear in the Euler-Poincaré reconstruction of the KdV equation, which suggests that they may be observable, for instance, through the Stokes drift of fluid particles in a shallow water channel.
