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

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. The partition function can include a linear statistic. The asymptotic formula involves a Fredholm determinant related to the Loewner energy of the curve, and also an expression involving the sampling function, the exterior conformal map for the curve and the Grunsky operator. The asymptotic formula also gives a central limit theorem for linear statistics of the particles in the gas.