An algebraic construction of a solution to the mean field equations on hyperelliptic curves and its adiabatic limit

In this paper, we give an algebraic construction of the solution to the following mean field equation:

$\Delta \psi +e^{\psi }=4\pi \sum _{i=1}^{2g+2}\delta _{P_{i}}$

on a genus $g\geq 2$ hyperelliptic curve $(X,ds^{2})$, where $ds^{2}$ is a canonical metric on $X$ and $\{P_{1},\cdots ,P_{2g+2}\}$ is the set of Weierstrass points on $X.$