An algebraic construction of a solution to the mean field equations on hyperelliptic Curves and its diabatic 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 $X.$ Furthermore, we study the rescaled equation $$ \Delta \psi+\gamma e^{\psi}=4\pi\gamma \sum_{i=1}^{2g+2}\delta_{P_{i}} $$ and its adiabatic limit at $\gamma=0$.