  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 $CP_{N-1}$ 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 $P(A-4\pi
N)=NT$ 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.
