Gauged vortices in a background

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 Kähler 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 exactly the partition function of the vortex gas interacting with a background potential. The coefficients of this analytic function provide geometrical data about each Kähler structure, the simplest of which being their symplectic volume (computed previously by Manton using an alternative argument). The interacting gas is found to obey a nonlinear deformation of Clausius’ equation of state. MSC (2000): 53C80, 37K65; PACS (2003): 11.27.+d, 74.25.Bt