Free energy of the Coulomb gas in the determinantal case on Riemann surfaces

We derive the asymptotic expansion of the partition function of a Coulomb gas system in the determinantal case on compact Riemann surfaces of any genus g. Our main tool is the bosonization formula relating the analytic torsion and geometric quantities including the Green functions appearing in the definition of this partition function. As a result, we prove the geometric version of the Zabrodin-Wiegmann conjecture in the determinantal case.

💡 Research Summary

The paper studies a Coulomb gas consisting of N charged particles on a compact Riemann surface M of arbitrary genus g, focusing on the determinantal case β = 1. The authors define the partition function Z_{ρ^{Ar},ρ^{can},N}(V) as an N‑fold integral over the surface, where the interaction kernel is the Green function of the scalar Laplacian and V is a quasi‑subharmonic external potential. Their main goal is to obtain a full asymptotic expansion of the free energy ln Z as N → ∞.

The analysis begins with a careful setup of the geometric background: a metric ρ, its associated volume form μ_ρ, the positive Laplacian Δ_ρ, and the scalar curvature R_ρ. The authors introduce three classical functionals—Liouville, Mabuchi, and Aubin‑Yau—that will later appear in the coefficients of the expansion. Two Green functions are considered: the standard Green function G_ρ and the Arakelov Green function G_{Ar,ρ}, each satisfying a specific normalization condition and having explicit transformation laws under conformal changes of the metric.

A key technical device is the bosonization formula, which relates the determinant of the magnetic Laplacian □{L,ρ^{Ar},h} on a positive line bundle L of degree k = N + g − 1 to a product involving the Arakelov Green function, a theta function, the determinant of the scalar Laplacian, and various geometric constants. The line bundle L is equipped with a Hermitian metric h, and the authors use results of Bismut‑Vasserot, Wentworth, and Wentworth‑Wentworth on the asymptotics of analytic torsion (the Ray‑Singer determinant) to expand det □{L,ρ^{Ar},h} in powers of N.

The computation proceeds in several steps. First, a “modified” partition function Z(θ) containing the theta function is evaluated using the bosonization identity. Then the average over the theta characteristics is performed, which eliminates the theta factor and yields the original partition function. The authors compute the difference between determinants corresponding to two different Hermitian metrics on L, and they expand the determinant of the scalar Laplacian det′Δ_{ρ^{can}} using known heat‑kernel asymptotics. The final result is the expansion

\