Geometry and Topology of Collective 2D Vortex Dynamics*

We consider the coarse-grained dynamics of many quantized chiral vortices on a 2D surface.

There are two groups of emergent phenomena. First, coupling the incompressible Euler equation to a metric in the presence of point vortices involves important ambiguities reminiscent of spin in quantum 2D systems. We obtain the coarse-grained flow of the vortex fluid, observe the effective negative temperature envisioned by Onsager in 1952, and discuss the curvature effects.

Secondly, vortices can form bounded drops of nonzero vortex density. The quantization of vortices leads to the presence of an "overshoot" in the vortex density at the boundary of the drop, also observed in Quantum Hall droplets. The Euler dynamics generates a remarkable edge wave in the overshoot density that obeys the integrable Benjamin-Ono equation. The mass of the soliton in this equation is quantized in units of one vorticity quantum, representing the topological nature of vortices.

Based on work with prof. Paul Wiegmann.