Thermodynamics of Vortices in the Plane

The thermodynamics of vortices in the critically coupled abelian Higgs model, defined on the plane, are investigated by placing N vortices in a region of the plane with periodic boundary conditions: a torus. It is noted that the moduli space for N vortices, which is the same as that of N indistinguishable points on a torus, fibrates into a C P N − 1 bundle over the Jacobi manifold of the torus. The volume of the moduli space is a product of the area of the base of this bundle and the volume of the fibre. These two values are determined by considering two 2-surfaces in the bundle corresponding to a rigid motion of a vortex configuration, and a motion around a fixed centre of mass. The partition function for the vortices is proportional to the volume of the moduli space, and the equation of state for the vortices is P ( A − 4 πN ) = NT in the thermodynamic limit, where P is the pressure, A the area of the region of the plane occupied by the vortices, and T the temperature. There is no phase transition.