Volume of Vortex Moduli Spaces

A gas of $N$ Bogomol'nyi vortices in the Abelian Higgs model is studied on a compact Riemann surface of genus $g$ and area $A$. The volume of the moduli space is computed and found to depend on $N, g$ and $A$, but not on other details of the shape of the surface. The volume is then used to find the thermodynamic partition function and it is shown that the thermodynamical properties of such a gas do not depend on the genus of the Riemann surface.