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

## TL;DR

This paper derives the asymptotic expansion of the Coulomb gas partition function on Riemann surfaces using bosonization, proving a geometric version of the Zabrodin-Wiegmann conjecture in the determinantal case.

## Contribution

It introduces a novel approach using bosonization to connect analytic torsion and geometric quantities for Coulomb gases on Riemann surfaces.

## Key findings

- Asymptotic expansion of the partition function derived
- Proves the geometric Zabrodin-Wiegmann conjecture in the determinantal case
- Establishes a link between analytic torsion and Green functions on Riemann surfaces

## Abstract

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.