Volume of Vortex Moduli Spaces

TL;DR

This paper calculates the volume of vortex moduli spaces on Riemann surfaces and demonstrates that the thermodynamic properties of a vortex gas depend only on the number of vortices, genus, and area, not on surface shape.

Contribution

It provides an explicit computation of the moduli space volume for vortices on Riemann surfaces and links this to thermodynamic properties, highlighting shape independence.

Findings

Volume depends on N, g, and A, but not on surface shape.

Thermodynamic properties are independent of the genus g.

Partition function derived from volume shows shape invariance.

Abstract

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.