Partition function for the 2d Coulomb gas on a Jordan curve

Abstract: We prove an asymptotic formula for the partition function of a 2d Coulomb gas at inverse temperature $\beta>0$ confined to lie on a Jordan curve. This also gives a central limit theorem for a linear statistic of the particles in the gas. We obtain different expressions for the asymptotic mean and variance which involve either the exterior conformal mapping of the curve or the Grunsky operator.