Gauged vortices in a background

TL;DR

This paper analyzes the statistical mechanics of gauged vortices on a sphere, using localization formulas to compute the partition function and derive thermodynamic properties, revealing geometric insights about the vortex moduli space.

Contribution

It introduces a method to explicitly compute the partition function of vortex gases on a sphere using localization formulas, linking thermodynamics with geometric structures.

Findings

Partition function computed explicitly for vortex gas on a sphere

Average height on the sphere derived from thermodynamics

Geometric data of Kaehler structures linked to physical properties

Abstract

We discuss the statistical mechanics of a gas of gauged vortices in the canonical formalism. At critical self-coupling, and for low temperatures, it has been argued that the configuration space for vortex dynamics in each topological class of the abelian Higgs model approximately truncates to a finite-dimensional moduli space with a Kaehler structure. For the case where the vortices live on a 2-sphere, we explain how localisation formulas on the moduli spaces can be used to compute explicitly the partition function of the vortex gas interacting with a background potential. The coefficients of this analytic function provide geometrical data about the Kaehler structures, the simplest of which being their symplectic volume (computed previously by Manton using an alternative argument). We use the partition function to deduce simple results on the thermodynamics of the vortex system; in particular, the average height on the sphere is computed and provides an interesting effective picture of the ground state.