Debye-Huckel theory for two-dimensional Coulomb systems living on a finite surface without boundaries

We study the statistical mechanics of a multicomponent two-dimensional Coulomb gas which lives on a finite surface without boundaries. We formulate the Debye--Huckel theory for such systems, which describes the low-coupling regime. There are several problems, which we address, to properly formulate the Debye--Huckel theory. These problems are related to the fact that the electric potential of a single charge cannot be defined on a finite surface without boundaries. One can only define properly the Coulomb potential created by a globally neutral system of charges. As an application of our formulation, we study, in the Debye--Huckel regime, the thermodynamics of a Coulomb gas living on a sphere of radius $R$. We find, in this example, that the grand potential (times the inverse temperature) has a universal finite-size correction $(1/3) \ln R$. We show that this result is more general: for any arbitrary finite geometry without boundaries, the grand potential has a finite-size correction $(\chi/6) \ln R$, with $\chi$ the Euler characteristic of the surface and $R^2$ its area.