A Quantum Hall Fluid of Vortices

In this note we demonstrate that vortices in a non-relativistic Chern-Simons theory form a quantum Hall fluid. We show that the vortex dynamics is controlled by the matrix mechanics previously proposed by Polychronakos as a description of the quantum Hall droplet. As the number of vortices becomes large, they fill the plane and a hydrodynamic treatment becomes possible, resulting in the non-commutative theory of Susskind. Key to the story is the recent D-brane realisation of vortices and their moduli spaces.