  Statistical mechanics of 1D multivalent Coulomb gas may be mapped onto
non-Hermitian quantum mechanics. We use this example to develop instanton
calculus on Riemann surfaces. Borrowing from the formalism developed in the
context of Seiberg-Witten duality, we treat momentum and coordinate as complex
variables. Constant energy manifolds are given by Riemann surfaces of genus
$g\geq 1$. The actions along principal cycles on these surfaces obey ODE in the
moduli space of the Riemann surface known as Picard-Fuchs equation. We derive
and solve Picard-Fuchs equations for Coulomb gases of various charge content.
Analysis of monodromies of these solutions around their singular points yields
semiclassical spectra as well as instanton effects such as Bloch bandwidth.
Both are shown to be in perfect agreement with numerical simulations.
